From the sounds of it, I suspect your MagSafe connects are dirty and/or the systems SMC services needs a reset.
Generally I don't recommend opening up the chargers as they are fuse welded and you often can't seal them back up as tight as they where after opening. Please be careful here as you can hurt your self or start a fire.
Try cleaning the contacts on both the charger and the system with a slightly damp cotton swab (Q-Tip) with a good grade of Isopropyl alcohol (85% or better).
Review this Apple TN on resetting the SMC services: Intel-based Macs - Resetting the System Management Controller (SMC). It will walk you thought the needed steps.
I would also download this little gem: coconutBattery. It will report to you what the SMC service of your system see's on the state of the battery and the charger. Post a screen shot here for us to see.
We also need to know what system you have. Can you give us the last four digits of your serial number so we can correctly identify your system.
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