Bevacizumab in Treating Patients With Relapsed or Refractory Chronic Lymphocytic Leukemia

Inclusion Criteria: * Diagnosis of B-cell chronic lymphocytic leukemia (CLL)\*, as defined by the following phenotypic characteristics: * Predominant population of cells share both B-cell antigens (CD19, CD20, or CD23) as well as the T-cell antigen (CD-5), in the absence of other pan-T-cell markers (CD-3, CD-2, etc.) * Mantle cell lymphoma must be excluded by demonstrating the absence of the t(11;14) by fluorescent in situ hybridization (FISH) * Dim surface immunoglobulin expression * Exclusively kappa and lambda light chains * Peripheral blood absolute lymphocyte count \> 5,000/mm\^3 * Lymphocytosis must consist of small to moderate size lymphocytes, with ≤ 55% prolymphocytes, atypical lymphocytes, or lymphoblasts morphologically * Requires chemotherapy, as indicated by any of the following: * Disease related symptoms, including the following: * Weight loss ≥ 10% within the previous 6 months * Extreme fatigue * Fevers \> 100.5°F for 2 weeks without evidence of infection * Night sweats without evidence of infection * Evidence of progressive marrow failure, as manifested by the development of or worsening anemia (hemoglobin ≤ 10 g/dL) and/or thrombocytopenia (platelet count ≤ 100,000/mm\^3) * Massive (i.e., \> 6 cm below left costal margin) or progressive splenomegaly * Measurable and progressive lymphadenopathy * Measurable (i.e., \> 5,000/mm\^3) and progressive lymphocytosis * Progressive disease or relapsed after or refractory to 1 course of an alkylating agent-based or purine nucleoside-based (e.g., fludarabine) regimen * No marrow function attributable to dysplasia related to prior therapy * ECOG performance status 0, 1, or 2 * Serum creatinine \< 2 mg/dL * If serum creatinine \> 1.5 mg/dL but \< 2 mg/dL, creatinine clearance must be ≥ 30 mL/min * Platelet count \> 30,000/mm\^3 * Direct bilirubin ≤ 2 mg/dL * Not pregnant or nursing * Negative pregnancy test * Fertile patients must use effective contraception * No other second malignancy within the past 2 years except squamous cell or basal cell carcinoma of the skin or in situ carcinoma of the cervix * No New York Heart Association class III or IV heart failure * No blood pressure \> 150/90 mm Hg * No unstable angina * No myocardial infarction or stroke within the past 6 months * No clinically significant peripheral vascular disease * No evidence of bleeding diathesis or coagulopathy * No significant traumatic injury within the past 28 days * Urine protein:creatinine (UPC) ratio ≤ 1.0 * Patients with a UPC ratio \> 1.0 must undergo a 23-hour urine collection and must demonstrate \< 1 gram of protein per day * No abdominal fistula, gastrointestinal perforation, or intra-abdominal abscess within the past 6 months * No serious, non-healing wound, ulcer, or bone fracture * No active infections requiring oral or intravenous antibiotics * No active bleeding or pathological conditions that carry a high risk of bleeding (e.g., known varices) * No thrombocytopenia requiring transfusion * See Disease Characteristics * More than 4 weeks since prior participation in an experimental drug study * At least 8 weeks since prior rituximab * At least 6 weeks since prior chemotherapy * More than 28 days since prior major surgery or open biopsy * More than 7 days since prior minor surgery, fine needle aspirations, or core biopsies * No concurrent major surgery * No concurrent participation in another experimental drug study * Concurrent full-dose warfarin or low molecular weight heparin allowed provided patient is on a stable dose AND INR is in range