Yes, one's mortgage holder is a key ingredient to how easy/quick things are.

We had to refinance anyway because our variable rate mortgage was running out and our mortgage company got all excited and said they could do it free and painlessly, no hoop jumping through (since they were already held our mortgage so had all the facts about us and our home) etc. and then the run around started. . . Several months later we were told we didn't qualify for that loan (that they had found for us, LOL. It was one of these new one's under Obama but we haven't ever been in "trouble" with our mortgage and weren't looking for relief of any sort, so it wasn't related to what we were looking for!) and they tried to keep the $300 they'd charged us!

We got the $300 back and learned they didn't really have any plans that would be a deal for us so we went looking elsewhere and found a great place; the entire thing was done online in about a week, very painless and the mortgage went up minimally (we'd switched types to fixed and the interest rate on this one was higher than our old 5-year rate that was running out).

Starting as soon as you can so you can take your time and work on making sure everything is right made it a bit less stressful for us. Even though the first try didn't pan out, nothing was lost because we still had plenty of time before our then mortgage would have run out. With the initial interview of several sources to see where to get the best deal, we got a sense of how our mortgage would change and what the issues/costs were so taking a bit of time until they went into effect helped us get use to the idea and shift our spending, etc.