Modern radar systems (note, not all of the radar systems currently being used even by the US military would be considered 'the most modern') should be able to handle a wide variety of targets at a variety of sizes, altitudes, ranges and speeds all at the same time. That being said, radars still fundamentally work by spewing radiation into specific regions of the sky - the more things that a radar needs to track, the more it needs to spread its power around.

Aside from the hard hardware limitations of the radar, what is more likely to be overwhelmed is either the attention and focus of human operators, or whatever software is being used to support the human operator.