I'm on my third Incredible (for various reasons). My most recent phone is having one of the same issues that my second phone had (but not the first).
I use Google Voice exclusively for calls and texts.
However, when someone sends me a text message, sometimes they show up right away, sometimes they show up a couple of hours later and occasionally they'll arrive as much as 12 hours later.
If I go in and refresh, they're there, but I don't want to keep refreshing all the time and I use texts for business, so they're often important.
In voice, I have synchronize inbox checked and background data enabled.
If a text is sent to my cell phone number it arrives instantly, so Verizon is telling me there's nothing they can do since it's a third party app that's causing the problem (Google Voice).