Our current loan is through Navy Federal Credit Union. The original process four years ago went very well, no complaints. NFCU can be fickle, though. We tried early this year to refinance our VA loan and were summarily rejected without a reason, early in the process; just a letter in the mail stating they couldn't offer us a loan. No one would pick up our calls or return our emails from then on...weird. A friend of ours had a similarly bad experience in that every time they called or received new estimates the numbers changed. He said they had to literally have them go through the number five times, and in the end, he just didn't feel comfortable moving forward.
Fast forward to today and they're willing to play ball and have been very attentive (likely because our house is worth substantially more than it was earlier this year). We did mention that we had quotes from other providers and were genuinely ready to jump on a much lower rate, which seemed to help. We sent them what Rocket Mortgage offered and they came very close, with lower closing costs. We just locked in a 2.411% APR (from 3.25% four years ago). We had an offer closer to 2.25% from Rocket Mortgage (rate, not including closing costs - I don't recall what the APR was). I tried to talk my wife into going with RM, but she refused if we could get even close with USAA or NFCU, which we did...so I conceded. We're still going to be saving around $200/month (keep in mind, we're resetting to 30 years from the 26 we still owed, so some of it isn't REAL savings. The real number, 26 mo. for 26 mo. is more like $180, IIRC (don't have the spread sheet at the moment))
The point: if you had issues refinancing earlier this year or last year now's the time to take advantage of rates before they go up. Who knows whether this housing bubble will be sustained, also.