In a block of non-volatile memory, bit line current increases with bit line voltage. For current sensing memory systems, average bit line current during a sensing operation need only exceed a certain threshold amount in order to produce a correct result. For the first word lines being programmed in a block, memory cells connected thereto see relatively low bit line resistances during verify operations. In the disclosed technology, verify operations are performed for these first programmed word lines with lower verify bit line voltages in order to reduce excess bit line current and save power. During read operations, this scheme can make threshold voltages of memory cells connected to the lower word lines appear lower. In order to compensate for this effect, various schemes are disclosed.