Bad PSU, video card or Ram are all a possibility so I suggest an inspection and cleaning of each, how to do this safely below;
If the computer needs cleaning out, first remove the power cord from the wall and open up the case, take anti static precautions before touching anything inside, you can do this by touching a bare metal part of the case, remove the video card, blow out the slot and re-seat the card securely, do the same with the memory sticks, clean all internal fans, if you use canned compressed air do not let any fan spin freely this can damage them, use something like a clean lollipop stick to hold the fan.
Make sure that all internal connections are secure and that there are no loose components, write down the PSU details (make and model) re-assemble the computer, boot up and see how it goes.
Just a cautionary note, if you do use canned compressed air do not upend the can as it is possible for fluid to be emitted.
Some information regarding PSUs;
As a PSU puts out various voltages +3.3V, +5V and +12V it may appear that the PSU is working correctly but it is not, any significant drop of any output can prevent the system from booting up, the other scenario is a significant increase in the output which can be worse as it can fry one or more major components such as the MB, CPU, Ram, add on video card etc.
Please be aware that there are no user replaceable parts in a PSU so a bad one should be disposed of in a responsible manner and any type of conclusive testing will need to be done by a suitably trained Tech who has the required testing equipment and the relevant knowledge as to how to use it.
Your upgrade options largely depend on your MB and in particular at the moment your video card and PSU, the ATI X1650 pro graphics card is available in both the PCI-E and AGP platform so we need to know which type your card is as you will have a lot more choice if the card is a PCI-E platform as the AGP slot type is basically a thing of the past and there are hundreds if not thousands of PCI-E cards available to choose from, even if the PSU is not bad it will more than likely need upgrading to power any new video card.