Is MD5 identical for corrupt/non-corrupt files?

I've been using the free Cobian Backup for about a year, with no problems. My backup procedure is to make full backups every 2 weeks, with daily incremental backups.

My 400GB external HDD is beginning to fill up. I need to remove some of the older full backups. Of course, the problem is that an older file may be non-corrupt, whereas a newer "copy" of the same file may have become corrupt.

I recently found another free tool, "Duplicate Cleaner". This file searches for duplicate files that can be identical in name, time, MD5 or any selected combination of these. I'm planning to use Duplicate Cleaner to remove duplicate files from my HDD.

So here is my question.

Is it possible for a corrupt file to show the same MD5 hash as an identical non-corrupt file? Or does the corruption produce a different MD5?

If the hashes are the same, then of course there is a danger that Duplicate Cleaner (or any tool) could delete the non-corrupt file and leave the corrupt file.

(Note that this thread is related to "MD5", rather than "Software".)
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