Vcc drops on connecting variable resistor between it and ground
I can't say what the cause is without more information, but since you ask "What could possibly be the cause of this?" I'll bite...

The cause is that more current is being drawn than the supply can give, so the voltage is drooping. You say that a "similar" board doesn't have the problem--do you mean a board with an identical schematic and layout? If so, the problem is less likely to be with the design, and more likely to be a flaw in the physical board or components. 

Start by checking the actual resistance across the resistor. Also, look for things like soldering problems that could cause a partial short between VDD and GND.