If they were Romanized Illyrians, I doubt they were looked down upon in Rome because of their origins. By the mid-3rd Century Rome had already had a couple of Romanized Celts, a Syrian, a few Africans, a Thracian, and an Arab emperor. At least some of these peoples would have been much more offensive than the Illyrians.
It's possible that the names of these emperors could give a clue to their origins - Claudius Gothicus and Probus both bore the praenomen/nomen Marcus Aurelius. This indicates that their families received citizenship possibly under M. Aurelius Antoninus, or more probably Caracalla. This in turn would imply that they were probably coming from a non-Roman, and maybe only recently Romanized background. But that's just my two cents.