Can an URI bring on a uri? I've had an URI for a few days, then saw the doctor yesterday for a painful uti. Initial urine workup showed a large amount of wbc's and blood in my urine. If both infections are bacterial in origin, did one bring on the other?

Unlikely. Uris are usually caused by viruses and utis are caused by bacteria.
Less likely. Upper respiratory infections tend to be caused by different bacteria than urinary tract infections due to the difference in the epithelium cells and their ability to adhere to it. Most uris are, however, caused by viruses and only a smaller portion is caused by bacteria, however, there are antibiotics that can kill both if both are bacterial, since some antibiotics have broad spectrum of action.