Mathematically it is quite feasible. Anyone who has taken a good multivariable calculus course knows (or knew) all of the basics necessary for moving an action game from 2 dimensions to 3, 4, or 17. It's certainly harder math, but not to the extent that it would be an obstacle. And since bots are essentially avatars controlled by real-time calculations involving the surrounding environment, controlling bots shouldn't be a problem either.
The problem would come with the human element. Creating artwork for the game that is meaningful in more than three dimensions might be tricky; how do you sculpt an n-dimensional model for a bot player if you don't even know how an n-dimensional human being ought to look? Assuming that problem can be circumvented (as well as the headache of designing aesthetically pleasing n-dimensional maps), the only real problem would be displaying the action on the screen in a comprehensible manner. As scienceman points out, the map would have the disconcerting behavior of pulsating and changing as your three-dimensional cross-sectional view changed orientation. And if one bot launched a rocket at another from a direction along which your cross-sectional view space was not aligned, the explosion would seemingly come out of nowhere.
But this all assumes that we are using a cross-sectional view space. Why not simply use a more traditional three-dimensional view and then use, say, color to indicate displacement along the fourth dimension? To visualize the concept, imagine an analogous situation in Flatland: A Square is crawling along his two dimensional floor when I fire a rocket at him from above and in front of him. He sees the rocket approaching as if it were in his plane, but it would be tinted bright red. As its vertical displacement decreases, the reddish hue would fade. When it is exactly level with him, it would be a neutral gray, and (assuming I missed), as it passed and continued downward beneath him, would fade to blue.
Of course, if the rocket isn't level with him, it shouldn't obstruct his vision. We would therefore paint it transparently until it crossed his plane. Walls, ceilings, and their fourth-dimensional equivalents could be treated in the same way for a three-dimensional player, and the entire view would be a shifting transparent haze of blue and red walls, players and rockets. It'd be confusing to look at but all the information would be there. Given sufficient time and the adaptability of the human brain (and a key to move ana and kata), I'd bet that a so-inclined player could even learn to play four-dimensional quake.
I don't really know how you'd tackle the fifth dimension, though, or even why you'd want to go as far as four. Three dimensions are still plenty for me.