That's actually pretty good. Double precision floating point (IEEE format, anyway), can represent all positive integers up to 9007199254740990, which takes you up to Fib(78), so the fact that you can get up to Fib(76) means you are getting close to the top. Good enough for most work.
After 9007199254740990, double precision only represents even integers. Fib(79) is odd, so can't be represented. Fib(80) would be representable though, as will be every other Fibonacci number for a while. Then comes a range where only multiples of 4 can be represented, and then a range with multiple of 8, and so on, so while there will be representable Fibonacci numbers in those ranges, more and more will be omitted.
After 9007199254740990, double precision only represents even integers. Fib(79) is odd, so can't be represented. Fib(80) would be representable though, as will be every other Fibonacci number for a while. Then comes a range where only multiples of 4 can be represented, and then a range with multiple of 8, and so on, so while there will be representable Fibonacci numbers in those ranges, more and more will be omitted.