Well, I determined that it was indeed the H: drive that was interferring/conflicting and causing the problem. All other computers in my department area actually had similar problems. It just depended on which order I plugged the 3rd device in, but whatever device was next in line to take "H:" as a drive letter assignment, it would have problems. I always thought it would just try to obtain the next letter (I: in this situation), but I guess not. I'm sure it has to do with the way the company drives are mapped or set up.
That being said, I decided to just call our tech support dept. I didn't really tell them the full reason since I wasn't sure how they would react to me wanting to plug in 3+ USB devices (the reason for the hesitation in the first place), but they actually had no problem sort of re-assigning my H: drive as a K: drive.
Actually, I should say more like "duplicating" cause what really happens now is that whenever I boot up any dept. PC, I now have BOTH a H: & K: drive that are mapped to the same location. I guess they essentially maybe added another script of some sort that points to the same spot.
To fully solve my dilemna, what I need to basically do is "disconnect" the H: drive if I want to hook up a USB device that will take that letter. The PC will then properly assign H: to a USB device. And the K: drive is still connected to the same source & all is good.