If the hospital tested me for a uti and it was negative would a badder infection show up on the same test?

Yes: urine examination would reveal bladder infection if present. The urinary track is composed of kidney, bladder and ureters. Sometimes it is necessary to culture the urine in the labs over a few days to see if any bacterial infections is present.

Should be: Urinary tract includes the kidneys, bladder, and the connecting ducts between them, and also the urethra, the duct from bladder to outside. Therefore, the tests for urinary tract infection (UTI) should be able to detect infection anywhere along the urinary system.