[remcob]

The way to combine hash functions for collision resistance is not composition (as with encryption) but concatenation: H'(file) = (H1(file), H2(file)). Now to have a collision on H' you need to collide both H1 and H2. But now pre-image resistance suffers.

[joveian]

Checking two full hashes and requiring both to match only improves pre-image resistance. However, you now need twice the space to store the hash and efficiency suffers, likely wose than the sum of the speeds due to cache effects of running two different algorithms on the data. If you use shorter or weaker hashes you might end up with two breakable hashes (either now or by some potential quantum computer) rather than one unbreakable hash.

Some package systems store multiple secure hashes and pick one at random to verify.