Can one discount code give different percentages at different cart values?

Yes. Create a discount rule containing one logic rule per tier. Each logic rule needs both its own outcome and its own condition, so that the outcome is bound to the threshold you intend. For example, one logic rule with a 20 percent item discount and the condition "cart over threshold" set to the higher amount, and a second logic rule with a 15 percent item discount and the condition set to the lower amount. Each outcomes needs pairing with its own condition.