If this is a carburetor model, you might also want to check the condition of your fuel pump - especially the diaphragm and gaskets - because if it would have developed a pin hole or something, it might not be getting quite enough suction when the tank gets emptier and it has to "pull" (suck) harder. With that same idea in mind, you might also want to check the fuel line for any place where a little air could be getting in around connections, or even small cracks in any hoses.
Checking the gauge and pick-up tube like Dan suggested is an excellent suggestion, and I'd suggest taking that even a step further to drain the tank to make sure there's nothing on the bottom like accumulated water (I would think a little less doubtful, but could cause major motor damage if it would happen to be), and it wouldn't be the first time some debris and/or hardware made it inside a tank and floats around until the level is low enough for it to get sucked against the pick-up tube to restrict the flow a bit.
One easy test (although I'd still recommend checking the other stuff regardless), would be if you have a portable tank (or know someone who might be willing to lend you one) that you could hook straight to the motor and then give it a try in similar conditions...it might confirm whether it's a fuel pump or other motor issue (possibly sticky carb floats at a high angle?) or whether it's a tank issue. Either way, you don't want to take a chance on running it "lean" for very long.
With any luck, it's something simple and not some combination or these or other things. Good luck!
