Why do we say that a good time response means a bad frequency response?
These terms refer to the general duality properties of these two domains. It should be obvious that higher frequency components mean faster changes in time and lower rise/fall times. In most natural systems, some of these fundamental properties oppose each other: Shorter rise time means more overshoot and higher settling times and vice versa; lower overshoot means a higher settling time as well as a higher damping factor resulting in a higher rise time. In general, achieving better frequency domain properties (in the sense of time domain duals) means worsening other time domain properties. Apart from that, in specific applications such as digital transmission, cleaner pulses (lower rise/fall time) means broader spectrum and frequency content, which is undesirable in the sense of bandwidth efficiency. But again there is no unique answer to what you have asked. Each application must consider its own tradeoffs.