I'm not sure about the specific details of the island example from above, but residency laws can get complicated. Summering for 3 months is obvious. But what about 5-4-3? What if you own property in one of the states? The list goes on. A given state will have laws specifying who can legally claim residency.

It gets really weird if multiple states attempt to "claim" you - I believe there was a supreme court case regarding state income tax in such a scenario.