Inclusion Criteria:
* Patient has signed the Informed Consent prior to any screening procedures being performed and is able to comply with the protocol requirements.
* Adults ≥ 18 years old
* Histologic or cytologic diagnosis of a WDNET, Ki67 ≤ 30%, unresectable, of foregut origin (thymic, bronchopulmonary, gastric, duodenal, and pancreatic) confirmed by the enrolling institution
\*Note: If patients have a functional NET, they are permitted to continue on a somatostatin analog for hormonal symptom control
* MSK patient has tissue available from a previous biopsy for the evaluation of potential predictive biomarkers. If tissue is not available for MSK patient, a new tumor specimen will need to be obtained prior to the start of study treatment If archived tissue is available, participating site patient will provide for the evaluation of potential predictive biomarkers. If tissue is not available for participating site patient, a new tumor specimen is optional prior to the start of study treatment.
* Documented radiological evidence for disease progression (measurable or nonmeasurable) ≤12 months prior to enrollment
* Disease that is currently not amenable to surgical resection with curative intent as determined by the treating investigator
* Measurable disease as defined by RECIST v1.1
* ECOG performance status 0 or 1 or KPS performance status 100 to 70
* Patient has adequate bone marrow and organ function as defined by the following laboratory values at screening:
* Absolute neutrophil count ≥1.5 x 10\^9/L
* Platelets ≥ 100 x 10\^9/L
* Hemoglobin ≥ 9.0 g/dL
* INR ≤ 1.5
* Serum creatinine \<1.5mg/dL or creatinine clearance ≥ 50 mL/min
* In the absence of liver metastases, alanine aminotransferase (ALT) and aspartate aminotransferase (AST) \<2.5 x ULN. If a patient has liver metastases, ALT and AST \<5 x ULN
* Total bilirubin \< ULN; or total bilirubin ≤3.0 x ULN or direct bilirubin ≤1.5 x ULN in patients with well-documented Gilbert"s Syndrome
* Negative serum pregnancy test done ≤14 days prior to registration, for women of childbearing potential only A serum pregnancy test will be conducted ≤ 72 hours prior to treatment start as a pre-treatment parameter. All women of reproductive potential and their partners must agree to use adequate methods of birth control (e.g. latex condoms) throughout the study and for 30 days after the last dose of study drug.
† A female of reproductive potential is a sexually mature female who: has not undergone a hysterectomy or bilateral oophorectomy; or has not been naturally postmenopausal for at least 24 consecutive months (i.e. has had menses at any time in the preceding 24 consecutive months).
* Patient with standard 12-lead ECG with the following parameters at screening (defined as the mean of the triplicate ECGs)
°QTcF interval at screening \<450msec (using Fridericia"s correction)
* Must be able to swallow LEE011 and everolimus capsules/tablets
* Recovered from adverse events (to grade 1 or less toxicity according to CTCAE 4.0) due to agents administered previously \*NOTE: Chemotherapy-induced alopecia and grade 2 neuropathy are acceptable
Exclusion Criteria:
* Patient has a known hypersensitivity to any of the excipients of LEE011 or everolimus
* Previous treatment with a CDK 4/6 inhibitor or an mTOR inhibitor
* Has had prior chemotherapy, targeted small molecule therapy within 2 weeks prior to study Day 1 or who has not recovered (i.e. ≤ Grade 1 or at baseline) from adverse events due to a previously administered agent
\*Note: Subjects with \< Grade 2 neuropathy or chemotherapy-induced alopecia are an exception to this criterion and may qualify for the study
* Patient has a concurrent malignancy or malignancy within 3 years prior to starting study drug, with the exception of adequately treated, basal or squamous cell carcinoma, non-melanomatous skin cancer or curatively resected cervical cancer
* Patients with central nervous system (CNS) involvement unless they meet ALL of the following criteria:
* At least 4 weeks from prior therapy completion (including radiation and/or surgery) to starting the study treatment
* Clinically stable CNS tumor at the time of screening and not receiving steroids and/or enzyme-inducing anti-epileptic medications for brain metastases
* Patient has impairment of gastrointestinal (GI) function or GI disease that may significantly alter the absorption of the study drugs (e.g., ulcerative diseases, uncontrolled nausea, vomiting, diarrhea, malabsorption syndrome, or small bowel resection)
* Patient has a known history of HIV infection (testing not mandatory)
* Patient has any other concurrent severe and/or uncontrolled medical condition that would, in the investigator"s judgment, cause unacceptable safety risks, contraindicate patient participation in the clinical study or compromise compliance with the protocol (e.g. chronic pancreatitis, active untreated or uncontrolled fungal, bacterial or viral infections, etc.).
* Uncontrolled diabetes mellitus as defined by HbA1c \>8% despite adequate therapy. Patients with a known history of impaired fasting glucose or diabetes mellitus (DM) may be included, however blood glucose and antidiabetic treatment must be monitored closely throughout the trial and adjusted as necessary
* Clinically significant, uncontrolled heart disease and/or cardiac repolarization abnormality, including any of the following:
* History of myocardial infarction (MI), angina pectoris, symptomatic pericarditis, or coronary artery bypass graft (CABG) within 6 months prior to study entry
* Documented cardiomyopathy
* Left Ventricular Ejection Fraction (LVEF) \< 50% as determined by Multiple Gated acquisition (MUGA) scan or echocardiogram (ECHO)
* Long QT syndrome or family history of idiopathic sudden death or congenital long QT syndrome, or any of the following:
* Risk factors for Torsades de Pointe (TdP) including uncorrected hypokalemia or hypomagnesemia, history of cardiac failure, or history of clinically significant/symptomatic bradycardia
* Concomitant medication(s) with a known risk to prolong the QT interval and/or known to cause Torsades de Pointe that cannot be discontinued or replaced by safe alternative medication (e.g. within 5 half-lives or 7 days prior to starting study drug)
* Inability to determine the QTcF interval
* Clinically significant cardiac arrhythmias (e.g., ventricular tachycardia), complete left bundle branch block, high-grade AV block (e.g., bifascicular block, Mobitz type II and third degree AV block)
* Patient is currently receiving any of the following medications and cannot be discontinued 7 days prior to starting study drug
* Known strong inducers or inhibitors of CYP3A4/5, including grapefruit, grapefruit hybrids, pummelos, star-fruit, and Seville oranges, that have a narrow therapeutic window and are predominantly metabolized through CYP3A4/5
* Herbal preparations/medications, dietary supplements
* Patient is currently receiving or has received systemic corticosteroids ≤2 weeks prior to starting study drug, or who have not fully recovered from side effects of such treatment
°The following uses of corticosteroids are permitted: single doses, topical applications (e.g., for rash), inhaled sprays (e.g., for obstructive airways diseases), eye drops or local injections (e.g., intra-articular)
* Participation in a prior investigational study within 30 days prior to enrollment or within 5 half-lives of the investigational product, whichever is longer
* Patient who has received radiotherapy ≤4 weeks or limited field radiation for palliation ≤2 weeks prior to starting study drug, and who has not recovered to grade 1 or better from related side effects of such therapy (exceptions include alopecia) or in whom ≥25% of the bone marrow (Ellis, 1961) was irradiated
* Patient has had major surgery within 14 days prior to starting study drug or has not recovered from major side effects (tumor biopsy is not considered as major surgery)
* Patient with a Child-Pugh score B or C
* Patient has a history of non-compliance to medical regimen or inability to grant consent
* Pregnant or nursing (lactating) women, where pregnancy is defined as the state of a female after conception until the termination of gestation, confirmed by a positive hCG laboratory test.