I was having the same problem with my Aspire 5750, purchased in late 2011. I did the following experiments before I've came to a fix:
1- Fix the charger itself, since it had a snappy cable (it almost broke). The problem got temporarily fixed and it's charging again, but the problem worsened a month after.
2- I sent the laptop then for a further investigation, a third party shop wanted to see if the issue was in the DC jack, but it was all fine. All they did was reapply a fix to the charger itself, again.
3- The problem is gone again, but is now back only a week later. I really got !#^&@@ off so all what was left for me to do is trying a static discharge. I got both the battery and charger off , pressed the power button for a whole minute (I know it's too much but I just wanted to make sure the mobo is entirely discharged), and I plugged everything back in. HOORAY! It works like a charm now: The power light never blinks and now my laptop won't shut down when the battery is taken off.
I'm not sure what exactly fixed the issue, but I can only guess a static discharge must be done often.