How's the battery life when you do unplug it? If it seems to be going through the battery quicker than normal, that and slow charging combined likely mean the battery is going bad.
What about tethering to a computer? If you do that and it seems to have trouble connecting, or giving you disconnect notices when you move the phone about or wiggling the cable, then it's likely a port related issue with the phone itself.
Sounds like it's safe to rule out the cable and charger as being the issue. If it's not the above symptoms, then my guess is a bad connection somewhere else in the charging circuit of the phone.