Vorinostat, Cladribine, and Rituximab in Treating Patients With Mantle Cell Lymphoma, Relapsed Chronic Lymphocytic Leukemia, or Relapsed B Cell Non-Hodgkin's Lymphoma

Inclusion Criteria: * Patients must be able to provide informed consent according to institutional guidelines * Patients must have: 1) MCL; or 2) relapsed or refractory cluster of differentiation (CD)20 positive B-cell indolent NHL; or 3) relapsed CLL * Patients must have measurable disease/disease status requirements as follows: * For CLL patients, symptomatic disease as defined by the International Workshop on Chronic Lymphocytic Leukemia (IWCLL) 2008 criteria that mandate treatment * For B-cell NHL patients must have at least one of the following to be eligible: * Positron emission tomography (PET) avid or measurable disease by computed tomography (CT) scan defined as at least 1 lesion that measures \> 2 cm in a single dimension * Significant bone marrow and/or peripheral blood involvement by NHL (i.e. leukemic phase) as determined by the investigator * Patients with Waldenström macroglobulinemia (WM) are exempt from this requirement if they have symptomatic hyperviscosity or clinically relevant cytopenias and elevated serum immunoglobulin M (IgM) * Patients must have adequate bone marrow reserve as indicated by an absolute neutrophil count (ANC) \> 1.500/mm\^3 and platelet count \> 150.000/mm\^3 if no bone marrow involvement; however, if there is significant lymphoma/leukemia bone marrow infiltration, no pre-existing hematologic parameters must be met * Patients must have a performance status of 0, 1, or 2 according to Eastern Cooperative Oncology Group * Serum creatinine \< 2.0 mg/dL or estimated glomerular filtration rate (GFR) \> 60 mL/min * Serum bilirubin =\< 1.5 × upper limit of normal (ULN) * Aspartate transaminase (AST)/alanine transaminase (ALT) =\< 2.5 × ULN * Alkaline phosphatase =\< 2.5 × ULN * Female patients of childbearing potential must have a negative serum pregnancy test within 2 weeks prior to enrollment * Male and female patients must agree to use an effective contraceptive method during the study and for a minimum of 6 months after study treatment Exclusion Criteria: * Significant hypersensitivity to cladribine or vorinostat; hypersensitivity to rituximab infusion is not an exclusion criterion; however, appropriate changes to infusion schedules will be made based on current or prior reactions * Current concomitant chemotherapy, radiation therapy, or immunotherapy other than as specified in the protocol * Patients with a diagnosis of a relapsed/refractory aggressive cluster of differentiation antigen 20 (CD20)+ B-cell neoplasm defined as Burkitt's lymphoma or diffuse large B-cell lymphoma * A diagnosis of acute lymphoplasmic leukemia, and lymphoblastic lymphoma * Use of investigational agents or any anticancer therapy within 2 weeks before study entry with the exception of hydroxyurea and steroids; the patient must have recovered from all acute toxicities from any previous therapy * Have any other severe concurrent disease, or have a history of serious organ dysfunction or disease involving the heart, kidney, liver, or other organ system that may place the patient at undue risk to undergo treatment * Patients with a systemic fungal, bacterial, viral, or other infection not controlled (defined as exhibiting ongoing signs/symptoms related to the infection and without improvement, despite appropriate antibiotics or other treatment) * Pregnant or lactating patients * Any significant concurrent disease, illness, or psychiatric disorder that would compromise patient safety or compliance, interfere with consent, study participation, follow up, or interpretation of study results * Patients with acquired immunodeficiency syndrome (AIDS) or human immunodeficiency virus (HIV) associated complex are not eligible for treatment * Patients with active hepatitis B or C are not eligible for the study * Patients taking other histone deacetylases (HDAC) inhibitors; for example, patients taking valproic acid, there must be a 14 day washout period prior to enrollment in this study