Brief Summary: This phase I trial studies the side effects and best dose of palbociclib with cisplatin or carboplatin in treating patients with solid tumors that have spread to other places and usually cannot be cured or controlled with treatment. Palbociclib may stop the growth of tumor cells by blocking some of the enzymes needed for cell growth. Drugs used in chemotherapy, such as cisplatin and carboplatin, work in different ways to stop the growth of tumor cells, either by killing the cells, by stopping them from dividing, or by stopping them from spreading. Giving palbociclib with cisplatin or carboplatin may help stop tumor growth in patients with advanced solid tumors.

Detailed Description: PRIMARY OBJECTIVES: I. Assess the safety and tolerability of palbociclib when administered along with cisplatin or carboplatin. II. Establish the recommended phase 2 dose (RP2D) of the tested combinations. SECONDARY OBJECTIVES: I. Characterize the pharmacokinetic (PK) profiles of cisplatin, carboplatin. II. Obtain preliminary evidence of anti-tumor efficacy of the tested combination regimens. III. Conduct PK/pharmacodynamics (PD) correlative analyses using palbociclib trough concentration and cyclin-dependent kinase 4 (CDK4) inhibition read-outs in tumor and surrogate samples collected on course 1 day 22 (C1D22). IV. Assess potential association between tissue-based biomarkers and efficacy. OUTLINE: This is a dose-escalation study. Patients are assigned to 1 of 2 arms. ARM A: Patients receive cisplatin intravenously (IV) over 30-60 minutes on day 1 and palbociclib orally (PO) once daily (QD) on days 2-22. Treatment repeats every 28 days for up to 6 courses in the absence of disease progression or unacceptable toxicity. ARM B: Patients receive carboplatin IV over 30-60 minutes on day 1 and palbociclib PO QD on days 2-22. Treatment repeats every 28 days for up to 6 courses in the absence of disease progression or unacceptable toxicity. After completion of study treatment, patients are followed for up to 4 weeks.