Sure, no problem. Most 529 savings plans have no state residency requirements. You can open accounts in as many of these states as you want, although in most cases there is little reason to have accounts in more than one or two states.

Can I contribute the maximum amount to each?

The IRS currently does not require that states count an investment in other state 529 plans when applying their own contribution limits. And there are no “contribution police” out there looking for people who are intent on using multiple states to stuff hundreds of thousands of dollars into 529 plans as a kind of tax shelter. But you are looking for trouble if you contribute more on an aggregate basis than you can reasonably argue might be needed for your beneficiary’s future higher education costs. Of course, between a pricey private college, medical school, and then business school you might be able to support the need to save a pretty hefty sum. However, a state will not want to see its program misused as a tax shelter (its tax status as a 529 plan could be threatened) and if a state determines that you have made contributions without the intent to use the account for college it will terminate your account and perhaps assess an extra penalty.