In this study we examine the role of inter- and intragrain current systems on the temperature-dependent critical current and magnetization of BSCCO-2223/Ag tapes. Experiments were done by measuring the sample magnetization field with movable miniature Hall sensors. We found that both systems have their own irreversible characteristics. The overall level of the intergrain (transport) critical current density, Jc, was limited at both lower (<30 K) and higher temperatures (30-95 K) by intergranular flux pinning, which seemed to be related to the maximum Josephson current of the weak links. Furthermore, we found that the irreversible intragranular magnetization has an influence on the magnetic field dependence of the intergrain Jc: when the intragrain pinning is strong (as at lower temperatures), the reduction of Jc in magnetic fields is moderate, and, when it is weaker (at higher temperatures), the decrease of Jc in fields is strong. This is a consequence of the field dependence of the maximum Josephson current density of the weak links, when the adjacent superconducting grains are in the mixed state.