I take the same stuff. Used to take its cousin Lyrica, but it cost so much I had to switch. The main difference is Neurontin upsets my tummy sometimes. So, I suppose it's possible you have a little bleeding in your tummy, your regular doc ought to figure out the source of your low white blood cell count, to eliminate that possibility. But I haven't had my blood count checked since I switched over to Neurontin.
By the way, those gaba drugs do have to be increased for a while, until it reaches a sort of plateau of effectiveness, so that's not unusual. I just don't know too much about how come blood cell counts change around, something about infection or bleeding. You could read the side effects of the medicine and see if it says anything about low white cells as a problem. But I don't think it would do that... but stuff like Prednisone will cause bleeding and thus a change in bloodwork labs. Sorry I can't provide more help.
I recently had an endescope done and there's no bleeding.
Your regular doctor ought to investigate the source of the low white blood cell count and come to some conclusion and treatment for it, if he thinks it's low enough to warrant checking it out. By the way, I've been taking gaba drugs for a couple years, and while I have not had any bloodwork done recently, I feel pretty good. So, consider how you think you are feeling these days... I mean, if you feel tired and flu-like, that might suggest you have an infection of some kind that might deplete your white blood cell count.
There is a cousin to Neurontin called Lyrica, much more expensive, it also needs to be increased as you go along until you plateau, but in general I liked it better mainly because it didn't make me as nauseus as Neurontin does. For overall pain and some pain that opiates won't help, those gaba drugs are miraculous, in my view. I hope you find out what is causing your low white blood cell count to drop, and also do check the list of side effects for Neurontin and see if it has any warnings about bloodwork results.
But I suspect your low white blood cell count is because of some other reason than the medication, becuz the only one I'm familiar with that CAN cause that is cortisone drugs. But as I said, I'm not a doctor, and definitely I'm not familiar with pharmaceuticals and their interaction and effects, other than from personal experience.