My experience was the the Electronic Systems lab (ELSYS), and it was initially a horrible experience. The main person coordinating the recruiting is an emeritus engineer, which means retired and therefore isn't in the office as much as a full-time employee. So e-mail replies were very few and far between. Luckily I had a personal referral so this might have expedited a bit. (as others have implied or stated, absolutely nothing happened with my online submission until I contacted someone directly)
For a few weeks I traded e-mails back and forth. Then, one Monday morning, I came in to my normal workplace to discover, in complete shock, an e-mail from a GTRI HR person which ended with "See you on Monday" a.k.a. that day, a.k.a. they COMPLETELY FORGOT to notify me of my interview!. Luckily I was able to rush home, throw a suit on, and head downtown and not be late...somehow.
It was a typical all-day interview for GTRI. I talked to many department heads and other senior engineers. Went out to lunch with the division chief and other senior personnel. Came back and did more of the same.
About a week later, I was notified by emeritus recruiter that I was no longer under consideration. It was quite disappointing, considering that one of the interviewing engineers told me (before I got the rejection) that he thought I'd be a great fit at ELSYS.
I e-mailed the two main people who had interviewed me, asking for constructive feedback. However, I received no replies back. If you have a high interest in working at GTRI (as I did), put up with it and try to find someone inside who you can contact. Otherwise if your interest in GTRI is passing, I would suggest that you not even bother.