You mean if they weren't a completely political organization that is incapable of taking on risky ventures without a firm dictate from above..
NASA has enough funding to do aggressive things if that was their goal, but they are more interested in pushing along everyone's pet project than they are in just achieving a goal.
If NASA was committed to going to Mars, they could do it, but not by fucking about with ion engines or nuclear rockets or building a spaceship in orbit.
Going to mars is risky, sure, but it does not require any technology we do not already possess. All you need to do is build a big fucking rocket(no pun intended) to lift a payload and throw it to Mars.
You don't need some new engine technology to shorten the trip to 30 days or some nonexistant EM shielding, you just need to be willing to say 'we are doing this because we consider it worth the risk'. I'd bet every single astronaut and astronaut candidate would gladly be willing to sign up for that mission. 1-2% increased lifetime chance of cancer to be the first humans on mars? Seems like a fair tradeoff to me.