Allied's not that great of a power supply brand from what I've heard. Can you tell us the model number or show us a photo of the label on the side of the power supply? Also, what case do you have? Some graphics cards can be quite long and may not fit in your case.
In general
the GTX 960 requires a 400~450 watt power supply, so if your PSU is capable of putting out that much wattage (on the 12V rail alone, not all the rails added together) then it should be alright. Otherwise you may want to get a new power supply.
For your price range, if you were looking for the best bang for your buck, the R9 290 is a better deal than the GTX 960 4GB.
They cost around the same as the GTX 960 4GB and also have 4GB of VRAM but absolutely beat the GTX 960 in terms of performance, being able to push framerates of up to 50% higher than what the GTX 960 is capable of. For example, you could spend
$243 after rebate on this Sapphire R9 290 Tri-X OC model and have enough left over to get a quality power supply, like this
Antec HCG 620 watt modular model for $45 after rebate or this
EVGA Supernova B2 750 watt modular model for $55 after rebate.
At the extreme end of your budget (
$300+), you may also want to look at the GTX 970.
Somewhat higher performance than the R9 290 but also runs cooler, quieter, and
uses less power than the R9 290. This might be the option you want if your Allied PSU turns out to produce enough wattage.