bad video card?

my niece has a dell desktop XPS_420 with an intel core 2 quad cpu Q6600 @ 2.40 ghz, 4.0 gb ram and windows 7 32 bit os. the video card is an ATI radeon hd 2600 XT. this computer is around 2 years old and it is one her husband use to use.
her computer will shut down after she has been on for awhile, just completely shut off. it happens mainly when she is online playing one of her games. i was wondering if the video card may be overheating and forcing the comp to shut down to protect itself. and if that is the case, what would be a reasonably priced video card to replace the one she has.
for the main game she plays, here are the system reqirements:
<table style="width: 648px;" class="stripesTable" border="1" cellspacing="1" cellpadding="1"><tbody><tr><th scope="row">Operating System</th><td style="text-align: center; vertical-align: middle;">Windows® XP/Vista/7 with latest service pack</td><td style="text-align: center; vertical-align: middle;">Mac OS® X 10.7.x (latest version)</td></tr><tr><th scope="row">Processor</th><td style="text-align: center; vertical-align: middle;">Intel® Pentium® D or AMD Athlon™ 64 X2</td><td style="text-align: center; vertical-align: middle;">Intel Core™ 2 Duo</td></tr><tr><th scope="row">Video</th><td style="text-align: center; vertical-align: middle;">NVIDIA® GeForce® 6800 or
ATI™ Radeon™ X1600 Pro (256 MB)</td><td style="text-align: center; vertical-align: middle;">NVIDIA GeForce 8600M GT or
ATI Radeon HD 2600</td></tr><tr><th scope="row">Memory</th><td style="text-align: center; vertical-align: middle;" colspan="2">2 GB RAM (1 GB Windows XP)</td></tr><tr><th scope="row">Storage</th><td style="text-align: center; vertical-align: middle;" colspan="2">25 GB available hard drive space</td></tr><tr><th scope="row">Internet</th><td style="text-align: center; vertical-align: middle;" colspan="2">Broadband internet connection</td></tr><tr><th scope="row">Media</th><td style="text-align: center; vertical-align: middle;" colspan="2">DVD-ROM drive required to install from disc</td></tr><tr><th scope="row">Input</th><td style="text-align: center; vertical-align: middle;" colspan="2">Keyboard and mouse required. Other input devices are not supported.</td></tr><tr><th scope="row">Resolution</th><td style="text-align: center; vertical-align: middle;" colspan="2">1024 x 768 minimum display resolution</td></tr></tbody></table>