Sometimes, they can't find a vein and suggest that some of them have actually collapsed. I'm usually well hydrated but on one occasion I was told to drink water before they tried again (it was in a US medical centre and did I take the mickey or what?)
I can't remember where I related this story before, probably on this forum but who knows. As kids we use to hit our hands with the teeth side of a comb, then swing our arms around until the mark made by the comb bled. No idea why we did it, we were kids. However it suggests that swinging your arm around increases the pressure of the blood in your arm and hand. I haven't thought to try it because it sounds daft but as I have some blood tests coming up soon, I might have to try it.