Assume you have two inputs, A and B.
A may hash to: A38uT75kjGz B may hash to: A38uHso629t
So yes, if you were to cut these off after the A38u then you would no longer be able to say for sure if you hashed A or B to arrive at your hash.
Of course in practice this usually isn't a problem as long as you have "a long enough" output.
Assuming a perfectly random uniform distribution, the usual desirable property of a cryptographic hash - the probability of a hash function not having a fixed point (that is, hashing at least one x to itself) is (1-1/n)**n, where n is the number of possible outputs. As n approaches infinity - which it does pretty rapidly in this case, since we're talking about 2**32 to 2**512 in practice - this approaches 1/e, or about 37%.
So, not only is it possible, but most "good" hash functions (63% of them) will have them.