I think grammar issues stem more from upbringing than education. You learn language by what you're around, so if everyone around you speaks a certain way, it's going to be natural to you to speak/write that way, as well. It's very difficult to untrain yourself from what sounds "right" to you. My grammar is not perfect but it's pretty good. It's not because I learned all the rules, but because what sounds "right" to me usually happens to also be conventionally correct. On the grammar portion of exams, I generally go with what sounds good, and I usually do pretty well.
However, if my parents, siblings, extended family, TV programs, etc. all used incorrect grammar, it would have been much more difficult to say/write things the correct way. Yet, I would still have the same intelligence and abilities to study for exams, answer test questions and I just get my papers checked for grammar, and voila! I'm a graduate.
I have a friend right now that owns his own mental health facility and is working on his second PhD, but when he speaks his ability to put sentences together correctly is still lacking. He can convey all of the principles and theories but subject-verb agreement ensnares him more times than not.
The sad thing is, people judge your intelligence by the quality of your grammar, so it can be a hindrance just like any other weakness, I suppose.
