I've played this game twice now, and both times I've been unable to max either paragon or renegade points.  This leaves some dialogue options greyed out.  The best example I can think of at the moment is influencing Jack after you've lost her loyalty to Miranda.  I made a concerted effort to get as many as possible, generally choosing paragon when a choice exists, and making sure to take whatever is available if only one choice appears.  
 
Am I missing something here? How do you max them out?

" I've played this game twice now, and both times I've been unable to max either paragon or renegade points.  This leaves some dialogue options greyed out.  The best example I can think of at the moment is influencing Jack after you've lost her loyalty to Miranda.  I made a concerted effort to get as many as possible, generally choosing paragon when a choice exists, and making sure to take whatever is available if only one choice appears.   Am I missing something here? How do you max them out? "

Forgot the name of it but isn't there a skill you can upgrade that increases your paragon and renegade points? Do that.

I never found a problem maxing out my paragon points. I even had some renegade points on my play through.

Huh, I hit all of the side missions I'm aware of. I didn't know about the multiplier skill, guess I'll figure that out. Sounds like the answer.

Talk to your party on your ship between missions. There will often be new dialog trees that lead to Paragon/Renegade points.

The multiplier skill is your character specific one (the one at the bottom) and yes, you need to do absolutely everything. Also its affected by the difficulty you are playing at.

First note that this is impossible unless you go all the way in one direction the WHOLE playthru. 
Also don't do any interruptions that are opposite your stance, sometimes this yields points.
There is a glitch on Samara's mission at the very end where you can ask her "Are you OK?" an infinite number of times but it would take forever.
 
If you have all but the last bar then u can get every dialogue line. (other than bringing someone back from becoming disloyal) 
Having that high will fix your loyalty issues before they happen & you won't need to MAX out. 
 
Keys are to wait  to do:
Both Jack and Miranda's loyalty (you can do one, but wait to do the other)
Wait to do Zaeed unless you go renegade.
Tali loyalty
 
Hope that helps. Speaking of impossible, getting the no one left behind on your first play comes to mind.

" I never found a problem maxing out my paragon points. I even had some renegade points on my play through. "

" @blacklab: The weapon and armors skill that you can evolve into Champion or Destroyer (?). The Champion one gives 100% Paragon/Renegade bonus, I think. "

You don't need 100% to convince Miranda or Jack in terms of the paragon/renegade choices, but I never found that to be an issue.

Some people seem to be putting in a lot of thought and work into this.  Maxing out one side is pretty easy and can be done long before the closing missions of the game. No need for special armor or anything.  On my last playthrough I maxed out the renegade side and was almost half way up the paragon side.  Maybe you aren't talking to your crew enough?  I'm not really sure where the trouble could be coming from.  
 
 Are you using the neutral response a lot?   It's not just about picking between the red or blue words.