My question is can a radio when turning up the bass push more than the set wattage on an amp. For instance if I have the Watts set to 250 on my amp will my radio push more if I turn up the bass on my radio. My speaker has an RMS of 250 W and I don't want it to push more if I Turn up the radio.
Your radio is not pushing any significant power. It just sends a signal to the amplifier. The amplifier then amplifies the signal, and by how much it does so depends on its gain setting. If you push a signal already has a great amplitude into an amplifier with the gain set too high, the amp will clip(distort) the signal because it doesn't have enough power to amplify the signal that much.
If you amp is rated at 500W it will push 500W without any clipping(if it is a good amp, that is). Since your speaker can only handle 250W, you'll need to stay way below the clipping/distortion point otherwise your sub will probably die. And beware that if you listen to bass-heavy tracks, even 250W may be dangerous after a while.
Thank you for the replies and I will keep that in mind! But one last question. Let's say my amp is set at 250 watts. If I turn the bass boost or sub level up on my amp will the amp then push higher than 250 even though it's set at 250?
No, it will just push more low frequencies at the expense of the mid range and high frequencies.
Back to the original question, standard output from something like a radio or CD player to an amp is typically less than 2 volts, with no current behind it, the amp will give you the 500watts when the input voltage is at its maximum, and the volume is at its maximum.
Since sound levels are logarithmic, you can still push that amp to about 8/10 on the volume without worrying about the speakers; I run a 100WPC amp into 20w speakers and hit 6/10 during parties