The ATX power supply is fairly standard - however there are 2 main standards, ATX and ATX12V (not counting AMD-approval on certain power supplies). The newer ATX12V standard increases power across the board (12V, 3,3V, 5V) to handle the demands of newer, high performance motherboard / CPU combinations (the higher the clock speed the more power it uses, all else being equal).
If your old power supply can't handle the demands of the new components then yes, your friend was right - the power supply might fail in time and possibly take out the motherboard. If you feel insecure (if your supply is, say, 250W or less) then for peace of mind swap to a higher, better quality one now - if only to sleep at nights, if it really bothers you. A new supply, especially ATX12V, will have better regulation of the 3.3V supply for the CPU (make sure it does). If over 250W it still depends on how much you have in the system. Your mileage may vary...