The strings tension will bend the neck a bit. I have a Rockbass and all I can say is don't be afraid to turn the truss rod. Until a few weeks ago my neck had some serious forward-bend, figured what the heck and started turning the truss rod pretty heavily. Now, the neck is straight (kinda). Depending on how your neck looks, i'd say a few millimeters wont cut it, certainly didn't for me.
This is my thoughts atleast, they might not be right but they worked for me. Not sure wether I should encourage you to do the same but it might work out.
