You risk overdrawing the PSU and damaging it. It can also cause other components to not get enough power or deal with erratic power from an overloaded PSU, so unrelated things may also start to act up. Better quality PSU's are probably going to be more robust than lower quality cheap OEM provided ones.
A video card or other component may be say it needs a 400w power supply, and it probably does only when fully loaded. So you may not experience problems on power on, but only after playing a game or something else that stresses the GPU.
Specific symptoms include:
display goes black
You can try "burning-in" your card by running a benchmark utility such as FurMark3d or similar for an extended time, giving problems a chance to surface. But don't do it if you know your PSU is underrated. Get a new PSU.