Inclusion Criteria:

* Diagnosis

* Pre-B ALL in first early (< 36 months from diagnosis) isolated bone marrow (BM) or combined BM/extramedullary relapse; or
* T-cell ALL in first isolated BM or combined relapse; or
* T-LL in first relapse
* Patients with leukemia must have had histologic verification of the malignancy at relapse, including immunophenotyping to confirm diagnosis
* Patients with lymphoblastic lymphoma must have measurable disease documented by clinical, radiographic, or histologic criteria; patients must have relapsed or become refractory to conventional therapy
* Patients must have a performance status corresponding to Eastern Cooperative Oncology Group (ECOG) scores of 0, 1 or 2; use Karnofsky for patients > 16 years of age and Lansky for patients =< 16 years of age
* Patients who relapse while receiving standard ALL maintenance chemotherapy will not be required to have a waiting period before entry onto this study
* Patients who relapse on therapy other than standard ALL maintenance therapy must have fully recovered from the acute toxic effects of all prior chemotherapy, immunotherapy, or radiotherapy prior to entering this study
* At least 14 days since the completion of cytotoxic therapy with the exception of hydroxyurea, which is permitted up to 24 hours prior to the start of protocol therapy
* At least 7 days since the completion of therapy with a biologic agent or donor lymphocyte infusions (DLI); for agents that have known adverse events occurring beyond 7 days after administration, this period must be extended beyond the time during which adverse events are known to occur
* No evidence of active graft-vs-host disease (GVHD) and >= 4 months must have elapsed; must not be receiving GVHD prophylaxis
* Creatinine clearance or radioisotope glomerular filtration rate (GFR) >= 70 mL/min/1.73 m\^2 or a serum creatinine based on age/gender as follows:

* 1 month to < 6 months (0.4 male, 0.4 female)
* 6 months to < 1 year (0.5 male, 0.5 female)
* 1 to < 2 years (0.6 male, 0.6 female)
* 2 to < 6 years (0.8 male, 0.8 female)
* 6 to < 10 years (1 male, 1 female)
* 10 to < 13 years (1.2 male, 1.2 female)
* 13 to < 16 years (1.5 male, 1.4 female)
* >= 16 years (1.7 male, 1.4 female)
* Total bilirubin =< 1.5 x upper limit of normal (ULN) for age
* Serum glutamate pyruvate transaminase (SGPT) (alanine aminotransferase \[ALT\]) < 3 x ULN for age, unless elevation due to leukemia infiltration
* Shortening fraction of >= 27% by echocardiogram, or
* Ejection fraction of >= 50% by gated radionuclide study
* No evidence of dyspnea at rest, no exercise intolerance, and a pulse oximetry >= 94% at sea level (> 90% if at high altitude)
* No evidence of acute pulmonary infiltrates on chest radiograph
* Patients with seizure disorder may be enrolled if on allowed anticonvulsants and well controlled; benzodiazepines and gabapentin are acceptable
* Central nervous system (CNS) toxicity =< grade 2
* Peripheral nervous system (PNS) toxicity < grade 3
* All patients and/or their parents or legal guardians must sign a written informed consent
* All institutional, FDA, and National Cancer Institute (NCI) requirements for human studies must be met

Exclusion Criteria:

* Patients with Philadelphia chromosome positive ALL are not eligible unless refractory to at least one tyrosine kinase inhibitor (TKI) therapy; patients that are unable to tolerate TKI therapy due to toxicity are eligible
* Patients with mature B-cell ALL, ie, leukemia with B-cell (soluble immunoglobulin \[sIg\] positive and kappa or lambda restricted positivity) ALL, with French-American-British (FAB) L3 morphology and/or a myc translocation, are not eligible
* Extramedullary disease status: patients with isolated CNS disease or isolated testicular disease are not eligible
* Patients with known optic nerve and/or retinal involvement are not eligible; patients presenting with visual disturbances should have an ophthalmological exam and, if indicated, an magnetic resonance imaging (MRI) to determine optic nerve or retinal involvement
* Patients with concomitant genetic syndrome: patients with Down syndrome, Fanconi anemia, Kostmann syndrome, Shwachman syndrome or any other known bone marrow failure syndrome are not eligible
* Cumulative prior anthracycline exposure must not exceed 400 mg/m\^2
* Patients taking anticonvulsants known to activate the cytochrome p450 system, in particular anticonvulsants such as phenytoin, carbamazepine, and phenobarbital, are not eligible; benzodiazepines and gabapentin are acceptable
* Patients who have previously received bortezomib or other proteasome inhibitors are not eligible
* Patients who have a known allergy to doxorubicin, cytarabine, both etoposide and etopophos, boron, mannitol or bortezomib are not eligible
* Patients who cannot receive any asparaginase products (E. Coli, PEG-asparaginase, or Erwinia asparaginase) on this study (eg, due to prior severe pancreatitis, stroke or other toxicity) are not eligible; patients who initially receive asparaginase, but must discontinue due to toxicity, remain eligible; patients with clinically significant prior allergies to pegaspargase are eligible if Erwinia L-asparaginase can be substituted
* Patients who are pregnant or breast-feeding are not eligible for this study; negative pregnancy tests must be obtained in girls who are post-menarchal; males or females of reproductive potential may not participate unless they have agreed to use an effective birth control method
* Patients must not have received any prior re-induction attempts and must not have received treatment for prior extramedullary relapse; patients with primary induction failure are not eligible