Welcome to Just Answer. I will do my best to fully answer your question(s).
There are a number of things that could cause the issue you are having with this toilet.
First, if there has been any replacement/ adjusting of the tank components the float and or fill valve could be closing/ shutting off prematurely to the full flush situation is begun. By "playing with" these areas ie. adjust flapper and or raise the water level in the tank this may solve your issue.
Now it may be that there is a partial clog in the toilet or drain line (this is my bet on this one). This is usually due to a large or thicker than normal paper usage. it can get stuck in the trapway of the toilet or hung up on some of the older piping. To correct this first try using a "closet auger" to rod out the toilet itself. They are for sale at any hardware store for around $20. If those helps dont fix the issue then its time to have someone rod out (snake) the older piping after removing the toilet. They also may want to check with a camera to see if anything else is going on inside the piping like a clogged vent, broken pipe ect. My bet is a partially clogged toilet trap that can be cleared witrh the closet auger though!
Thanks for the guidance; I will buy and try the closet auger and I will get back to you.
Thought I'd follow up: first bought a closet auger: no luck. Then purchased a 3/8" 50 foot snake, dropped it in the vent pipe, and found no obstructions. I thought I'd also offer: 3 months (Nov 2007) after we moved back in after the remodel (Aug 2007), the lateral backed up and drain water was emerging thru this bathroom's tub, adjacent to the toilet. So this bathroom seems to be 'ground zero' for plumbing problems...
Still, tho: toilet flushes reliably with a full tank, but not with a partial tank. Also, in regard to toilet tank hardware (float, etc.): we've never changed anything and the sister toilet is fine.