As others have pointed out, it may well be a packing issue.
But if you literally can't light the tobacco, then I think you may have a poorly drilled pipe that doesn't position the draft hole close enough to the bottom of the bowl. You can check this out by running a clean white pipe cleaner through the stem and into the empty bowl. Shine a light inside and see how high up the bowl the draft hole is. If it's too high, you'll never be able to finish a bowl.
OTOH, it's also possible that your tobacco has been collecting moisture throughout the smoke and is now simply too wet to light. (When you dump out the remnants, are they dry, unburnt strands of tobacco or are they dark and wet?) If that's the case, you'll likely hear gurgling as you puff. When that happens, run a pipe cleaner through -- while you're smoking -- to soak up some of the moisture. Some tobaccos naturally burn a bit "wetter" than others, but heavily cased aromatics are probably the worst culprits. So you may want to experiment with different blends and see if that makes a difference.
Moisture can also build up quickly if you're puffing too fast, something most of us have had to learn to correct over time.
Having said all that: Yes, I have to agree with those who have pointed out that you don't get an award for completely finishing every bowl. If it's starting to taste bad, by all means stop. Even so: if you're consistently dumping out unburnt tobacco -- and especially if you're doing it regardless of the blend you're smoking or the pipe you're smoking -- it could be a symptom of a poorly made pipe, an overly wet tobacco, or issues with your packing technique and/or smoking speed. I wouldn't just ignore those symptoms if you're wanting to improve in the art of pipe smoking.
Finally, I'll just add that I've never dumped ash in the middle of a bowl. The top layer of ash actually helps to cool the burn -- the same principle behind trying to maintain a half-inch or so of ash on a fine cigar -- and I also believe it creates a protective layer that encourages combustion beneath.
Bob