We have measured and analyzed both the static and dynamic contribution to the anomalous nonmonotonic dependence of the irreversible magnetization, M(B), for underdoped ${\mathrm{Bi}}_{2}$${\mathrm{Sr}}_{2}$${\mathrm{CaCu}}_{2}$${\mathrm{O}}_{8+\mathit{y}}$ and overdoped (${\mathrm{Bi}}_{1.6}$${\mathrm{Pb}}_{0.4}$)${\mathrm{Sr}}_{2}$${\mathrm{CaCu}}_{2}$${\mathrm{O}}_{8+\mathit{y}}$ single crystals. We extracted the ``unrelaxed'' magnetization ${\mathit{M}}_{\mathit{c}}$(B) and traced the time evolution of M(B,t) from short to long time scales using combined magnetization and flux-creep measurements. Despite their large differences in anisotropy, pinning force, and flux-creep rate s(B)=dln M/dln t, both samples showed the anomaly over a wide range of T, the anomaly being most pronounced in the ``unrelaxed'' state. This suggests that the primary cause of the anomaly lies in the flux-pinning defect structure of the material. However, M(B,E) also essentially depends on the induced electric field E in the sample, so the shape of the M(B) curve can strongly change with time due to the relaxation of E(t) and the nonmonotonic dependence of s(B).