Can an mbed device be damaged by something drawing too much current?
The mbed wiki shows has a (partial) schematic of the mbed module. It is using a LD1117S33 3.3v regulator.

This is a 800mA regulator, but it is likely that the mbed's implementation is good for significantly less than that, given the apparent lack of any heatsinking on the voltage regulator.

However, the LD1117 does have internal thermal and current limiting, so it should be pretty bulletproof with regard to damaging it.

Basically, if you draw too much current, the regulator will get hot, the thermal limiting will kick in, and it will reduce the output voltage until it no longer overheats. This could cause unexpected/strange behavior in the LPC MCU (unless it has brownout detection), but it shouldn't damage anything.

Basically, you don't need to worry about drawing too much current.