The existence of a remanent magnetization (M_rem) on switching off the field of a field cooled (FC) sample of a highT_c superconductor is often reported. It has recently been argued thatM_rem should equal the difference in FC and zero field cooled (ZFC) magnetizations (M_FC —M_ZFC) in hard superconductors and this has been demonstrated to hold in single crystals of YBCO at 4.2K over a limited range ofH values. We report the detailed magnetization measurements under various thermomagnetic histories (of whichM_rem is one special case) on two specimens of Nb, which show different extents of flux trapping. We find that there are in general three regions inH, T space, corresponding toM_rem+M_ZFC−M_FC=0,M_rem<(M_FC−M_ZFC) andM_rem>(M_FC−M_ZFC). At anyT, the equality holds forH<H_c1(T), and forH→H_c2 (M_FC−M_ZFC) asymptotically vanishes and thereM_rem>(M_FC−M_ZFC). We show that there exists an intermediate region in all hard superconductors, whereM_rem<(M_FC−M_ZFC). The range over which this situation persists, however, depends on the degree of irreversibility in a sample. We can explain qualitatively all the history dependent magnetization data in terms of the critical state model. We point out an inconsistency in an earlier analysis to determineH_c1(T) from such data in YBCO. We also propose a new criterion for putting limits onH_c1(T) in hard superconductors.