Why does Number() return wrong values with very large integers?

Javascript uses 64-bit IEEE-754 binary floating point to store all numbers - like double in C# and Java, for example. There isn't a different type to store integers. (The actual implementation may use optimizations to avoid always performing arithmetic in this way, but from an end-user perspective the results will always be as if every number were treated as a 64-bit binary floating point value.)

That means only 52 bits are available to store the significand, with the other bits being used for the exponent and sign. With normalization, that means you can effectively store values with 53 significant bits of precision. That means beyond 2⁵³-1 (which is the value 9007199254740991 as quoted in other answers), the distance between "adjacent" numbers is more than 1, so you can't store all integers exactly.