As it turns out, of the students in the three classes I've studied, the procrastinators didn't write particularly worse papers. At least not by the two indicators I've looked at so far.
One might think that waiting to the last minute to start a research paper would result in it being too short. Nope. There was more difference in length from one class to another than between procrastinators and non-procrastinators. One of the procrastinators actually wrote the longest paper in his class. None of them wrote the shortest. Procrastination has no clear consequence on paper length.
It would also seem reasonable to expect that procrastinated papers would contain more errors, for two reasons. One, waiting too long to start the paper may leave the writer without time to proofread. This should leave evidence of "performance errors"--the kinds of mistakes we all make when we don't watch ourselves. Two, students who avoid the task of writing may well be doing so because they believe they are bad at it. These students might leave evidence of "cognitive errors"--the kinds of mistakes we make when we are confused about the rules. These mistakes will simply not be caught by self-proofreading and can actually be made worse (since the proofreading is done by the student's wrong impression of the actual rules).
But no, procrastinators don't make more errors. Of both classes with procrastinated papers, the error rate was lower with procrastinators. The procrastinators in one class had an error rate of .5 errors per 100 words. The class average (median) was .67--slightly higher. In the other class, procrastinators had an error rate of .9 errors per 100 words. That classes averaged 1.7.
Here again the difference between classes is stronger than the difference between procrastinators and students as a whole. Procrastination doesn't increase errors, but the numbers are awfully close (.5 to .67 and .9 to 1.7) with a limited amount of data (less than fifty students) to jump to the conclusion that it helps.