The best they could probably do would be a lawsuit, but the states would be well within their rights to add new laws on top of federal ones.
The FCC first tried to regulate net neutrality, but that was struck down by a court as beyond its remit, since ISPs weren't common carriers. So the FCC declared that ISPs were in fact eligible for Title II regulation, which effects preemption of state-level regulations. Except the current administration has had the FCC declare that ISPs are not Title II carriers, and therefore FCC has no right to regulate them... which makes it difficult to argue that the federal regulations (which no longer exist) can preempt state regulations.
Cable ISPs weren't/aren't.
Dial-up ISPs have been under Title II since the 90s.
That is, net neutrality applies if you access the internet via dial-up, but not via cable/fiber.
How so?
Based on what I've read, I've gotten the impression that the Internet will be primarily regulated by the FTC, instead of the FCC. That would mean that ISPs aren't able to engage in anticompetitive practices, but are pretty free otherwise.
I haven't heard anything about states' standing on the issue. Unless there are explicit federal laws saying that the government can't regulate the Internet (which seems unlikely), it's pretty cut-and-dried that states would be able to pass legislation to affect ISPs in their jurisdiction.
Sorry if my take is a little confusing, here is some more coverage: [2], [3].
[1] http://www.opn.ca6.uscourts.gov/opinions.pdf/16a0189p-06.pdf
[2] https://www.forbes.com/sites/danielfisher/2016/08/10/fcc-los...
[3] https://arstechnica.com/tech-policy/2016/08/in-blow-to-muni-...
A law banning NN will never get passed due to gridlock (thankfully), so barring some huge change in public opinion I don't see states being prohibited from enforcing NN.
It's going to be interesting to see if the FCC tries to strike those local laws down and how ISPs will handle the things from State to state.