I think a reasonable example is Lamé's proof of Fermat's theorem. Experimental evidence confirmed that Fermat's theorem holds for small numerical examples, and Lamé's proof shows that if certain rings are unique factorization domains (which is a similar assumption to Euler's assumption that the product formula for sine holds) then Fermat's theorem always holds. Unfortunately, the unique factorization assumption sometimes fails.

Granted, the example isn't perfect because Fermat's theorem did turn out to be true, just not for the same reasons.