A non-volatile storage system is disclosed that includes pairs of NAND strings (or other groupings of memory cells) in the same block being connected to and sharing a common bit line. To operate the system, two selection lines are used so that the NAND strings (or other groupings of memory cells) sharing a bit line can be selected at the block level. Both selection lines are connected to a selection gate for each of the NAND strings (or other groupings of memory cells) sharing the bit line. One set of embodiments avoid unwanted boosting during read operations by keeping the channels of the memory cells connected to word lines on the drain side of the selected word line biased at a fixed potential.