Next thing i'm going to try is my backup Radeon 260X which requires a 6-pin PCI-E connector.. Obviously which the Dell PSU does not provide.. But I have a Sata to 6-pin PCI-E adapter to provide power being shipped to me tomorrow and will let you know if this works.. I'm certain it won't work either as the 260X has stronger PSU requirements than the 750 Ti does but I will update and let everyone know.
As someone who's tried both cards in a HTPC (perfectly with a 360w PSU), the 260X draws roughly 20-30w more power under load and is about 10% slower. I'm wondering if your problem is more your motherboard as merely booting a 750Ti won't draw anywhere near its maximum power. However, if the motherboard is not providing enough juice because its VRM's / power circuitry has been "designed down" to just what it came with, then it may well fail no matter what PSU you get. You could always try a 750Ti with external 6-pin connector (eg, Asus twin-fan non-Strix variant GTX750TI-OC-2GD5).
If you want a real world example of the 750Ti, the card itself barely pulls 74w in Furmark, nearer 50-60w in most games (that's 5-7 amps at 12v). "At the wall" whole system power draw is often 120w when gaming (just about maxed out with Skyrim on Ultra + 2K HD texture pack) even with an i5-3570. Lighter games + VSync on = under 100w. In short, you've either got a very flaky PSU or a very flaky motherboard, as some people are successfully using the 750Ti even on 180-200w SFX PSU's in Mini-ITX cases.