Bent bar probably won't throw the cut off to one side but it is quite common for one rail to wear down more than the other so the chain doesn't sit flat. Usual cause is not cleaning out the oil hole through the bar, so one side get starved of oil and wears more. If its not too bad most dealers can grind the rails off level.
Does putting the bar on the other way up cure the problem? That would indicate that it is the source of the trouble, provided its not uneven both sides. Just re read the OP, cutting OK with the tip but not the near part does sound very much like the bar is the problem.
Other possible causes:
Underside of the chain is worn more one side than the other - same cause as the above, only answer is a new chain but DON'T put it on before getting the bar trued up.
Depth gauges [raker things on front of the cutters] higher one side than the other - you can get filing guides / gauges to check that.
You can get gauges to measure the teeth and check the angles are the same each side, if they aren't either correct it by sharpening or take it into a shop to get it ground on a machine. Also check the vertical angle you hold the file [usually should be level or handle end 5 degs down. Holding it higher one side than the other will also throw things out. However minor discrepancies won't make a massive difference.