I. To determine the optimal biologic dose (OBD) for poly (adenosine diphosphate [ADP]-ribose) polymerase (PARP) inhibition using irinotecan (irinotecan hydrochloride) (once weekly intravenously in 2 of 3 weeks), in combination with ABT-888 (veliparib) (twice daily orally for 2 of 3 weeks). (ORIGINAL DOSE ESCALATION PORTION) II. To determine the recommended phase II dose (RP2D) for irinotecan (once weekly intravenously in 2 of 3 weeks), in combination with ABT-888 (twice daily orally for 2 of 3 weeks), determined by evaluating the feasibility, safety, dose limiting toxicities and the maximally tolerated dose. (ORIGINAL DOSE ESCALATION PORTION) III. To determine the safety profile of irinotecan in combination with ABT-888: the incidence of adverse events (AEs) and clinically significant changes in laboratory tests, electrocardiograms (ECGs), and vital signs. (ORIGINAL DOSE ESCALATION PORTION) IV. To determine the safety profile of irinotecan in combination with ABT-888 at the recommended phase II dose: the incidence of adverse events (AEs) and clinically significant changes in laboratory tests, ECGs, and vital signs. (ORIGINAL DOSE ESCALATION PORTION) V. To determine the recommended phase II dose (RP2D) of each drug for irinotecan (once weekly intravenously in 2 of 3 weeks), in combination with ABT-888 (twice daily orally for intermittent dosing days 1 to 4 and days 8 to 11 of each cycle), determined by evaluating the feasibility, safety, dose limiting toxicities and the maximally tolerated dose (MTD). (DOSE ESCALATION FOR INTERMITTENT ABT-888 PORTION) VI. To determine the safety profile of irinotecan in combination with ABT-888: the incidence of adverse events (AEs) and clinically significant changes in laboratory tests, and vital signs. (DOSE ESCALATION FOR INTERMITTENT ABT-888 PORTION) VII. To determine the safety profile of irinotecan in combination with ABT-888 at the recommended phase II dose: the incidence of adverse events (AEs) and clinically significant changes in laboratory tests, and vital signs. (DOSE ESCALATION FOR INTERMITTENT ABT-888 PORTION)