Can insecure algorithms be combined to form a secure algorithm? This is hypothetical as I can't think of any reason to do this, but out of curiosity...

Could I, for example, take the MD5 digest of a message and concatenate it with the SHA-1 digest (not quite broken, but getting close), to form a secure concatenated digest? Intuitively, I would think the chances of finding a collision that crossed the two algorithms would be low enough to make this secure, but I'm not sure how to verify that.
At the time of answering, the question was can insecure algorithms be combined to form a secure algorithm?



Yes: Think of any secure round-based block cipher. The independent rounds are not secure, but put together the overall cipher is. I think my favourite example is (currently) the Even Mansour cipher,
which combines two xor operations and one unkeyed permutation to form a secure cipher.

 nb: I know this isn't a direct link, but its the best introduction I know of

For a good discussion of the hash example in your question, read this question.