I would be forthcoming if I were you. SInce it was your very first job, it would not be hard to surface if a thorough background check was done. However, I would take a more proactive view of the situation. by expressing your desire to move past the new-grad category as quickly and as SAFELY as possible.
That being said, your situation, based on MY assumption without all the details of your time spent passing medications, is one MAJOR pet-peeve of mine. If the manager and preceptor had an issue with your timeline, I venture to guess that there were other tasks and possibly more patients involved than a preceptor should have had that allowed for some things to get missed. Had you been in training during the days when disturbing the medication nurse was practically a firing offense, you would probably be well on your way to an insurmountable confidence level. But as luck would have it, you preceptor probably had more added onto her plate than she could possibly handle and you, Mr/Miss Newcomber, had to be the head that rolled. Unless there is a profound disregard for patient safety, co-workers, and the job itself, NO ONE should be fired within weeks of entering a brand new career field. They showed that they were no better than the ones demanding the one-year experience before the ink dries on your license! I mean, 4 weeks???? Sounds like my experience in my attempt in learning Spanish. I sat down at my desk on the first day of class. The instructor said, "fakejrhfbaeliufbaliufhEachKearjbflhjefmsgtiwwtywn tkuurjtrayki, si?" So I got up and left!!!