The probability of a hash collision between n files if there are N possible hashes, assuming that each hash is equally likely, is roughly 1-e^(-n^2/(2N))[1]. Let's suppose that there are roughly a trillion, that is, 2^20, files; this is way more than the actual number of files, by the way. And let's suppose that we have a 256-bit hash such that every 256-bit string is equally likely(a reasonable assumption as long as the hash hasn't been defeated)--the number of possible hashes is 2^256. So the chance of a collision is roughly 1-e^(-2^80/(2^257)); since the number inside the exponential is so small, we can approximate it as 2^-177, or less than 10^-53. That number is so small that it's more likely your office will get hit by multiple independent meteors.

[1]This formula is taken from http://en.wikipedia.org/wiki/Birthday_paradox