This is a construction that is restricted to certain dialects of US English.
It is not ungrammatical in the sense of syntactic well-formedness; that is, when someone says a sentence like this they are not making a speech error. Instead, this construction is simply non-standard, which means it is something to avoid in a formal situation. Informally, this is used throughout the southern US, but has not spread to any other region I am aware of. Interestingly, it so happens that the same construction is standard in German.
What is going on in "might could" constructions is a process called "modal stacking", where multiple modal verbs (e.g. "could", "should", "might", "would", etc.) can be stacked on top of each other. Each added modal verb contributes towards the overall meaning of the sentence. In Standard English, to convey the same meaning, we have to use another construction:
I might could do that. --> I might be able to do that.
We are doing effectively the same thing in standard English in terms of semantics, it's just that we have to change things around to get around a syntactic restriction.
Contrary to what waiwai933 says, these constructions are not redundant by definition (they are only redundant if you stack them redundantly!). Neither "I might do that" nor "I could do that" would have the same meaning as "I might could do that".
Other constructions include:
- I might should do that. (= "Maybe I should do that")
- I used to could do that. (= "I used to be able to do that")
Isn't sloppy, meaningless, or redundant; linguistically, it is a systematic process (which I think is really cool!). It is just non-standard in English — something one would probably avoid in formal situations, just like contractions in writing, double negatives, slang words, etc.