Will a bad neuteral switch cause transmission not to shift into overdrive?

Yes, a faulty neutral safety switch can prevent a transmission from shifting into overdrive. The neutral safety switch ensures that the vehicle is in the correct gear for safe starting and shifting; if it's malfunctioning, it may incorrectly signal the transmission control module. This can lead to issues with shifting, including not engaging overdrive. It's advisable to have the switch inspected and replaced if necessary to resolve shifting problems.