In Canada, the single payer healthcare system is Provincial, not Federal, and yet the income tax brackets at Canada's Federal level are roughly equal to those in the US (in fact are actually higher for some earners).

If we wanted to take Canada's cue, there's plenty of room to increase State tax rates even if we did nothing about Federal taxation. Or we could take Switzerland's cue and totally flip the distribution, where Cantonal taxes far exceed Federal income tax.