I see problems like this very frequently, usually with low-end OEM PC's that the user has gotten disgusted with the crappy performance or lack of installed features and decide to "upgrade the old girl". PC store salesmen rarely, if ever, inquire, advise or are even aware of the PSU capacity of the customers computer; they're just happy to send you out the door with a new fancy graphics card, a dual-layer DVDRW, an extra stick of RAM or two and maybe a second or larger hard drive to boot. Double the risk if the computer user decides to light up his or her case like the Las Vegas Strip. Rarely, if ever, is any thought given to the the rating of the PSU during the DIY upgrade process. For that matter, we've seen customers computers that were upgraded by PC technicians who ought to have known better but obviously didn't.
Given a constant mains power, more load means more current draw. More current draw means more heat. The PSU struggles to keep up and the voltage on the rails drop. Less voltage with the same load means even more current and more heat until something gives. If you're lucky, it'll ONLY be the PSU, but we've seen fried CPUs, Motherboards, Memory, individual cards, all damaged due to an overloaded PSU.
Most low-end OEM machines do not have large or very high quality power supplies. They are designed to handle the load of the machine (or at least we hope so) as it originally came from the factory and not much more. At $299 for the weekend or holiday special from Dell (and a free printer besides!), how much PSU quality do you think you're really getting?
Now that I'm done ranting, to answer your specific question, I would strongly advise you to replace your PSU with a good decent-quality unit with a rating somewhere between 300-350 Watts. You have to bear in mind, as Bob alluded to, that PSU units often do not live up to their billing, and often are rated at peak load, not constant load. Thus, your current 250W PSU may only be putting out 150-200W under constand load.