"Flat" means a component or speaker is not likely to color the sound on its own. The problem is that there are many different components in the system...and then there’s the room itself...and your own hearing, your mood, etc.
The problem is not so much the individual component or speaker. The problem is in trying to get all the different freq. curves (each with their own slight variations) to work together and add up to one final in-room response that you can live with...flat or otherwise.
The big lie is to think that because you’ve bought a component or a speaker with a "flat" response (or in fact any stated response) that therefore you will be assured that it will remain so in your system and in your room. It isn’t so. The room may dominate, other components may dominate. Bad interaction with certain product combinations may dominate. Even things like a lack of power-conditioning may dominate. All you have bought is a component that will ’allow’ for a flat response under certain conditions (sometimes those certain conditions are specified and sometimes they are not), but the flatness is not guaranteed in your setup, only on the test equipment used to test that piece of gear. But, "most" gear is designed to be used successfully in "most" rooms and with "most" other gear.
A truly flat response may actually be audibly (subjectively) desirable, but almost always only with an Extremely well-executed system of rather expensive, high quality and full-range gear. In fact, it might surprise a lot of people just how much attention to detail and how much money must be spent to subjectively feel safe from any real concern with frequency response issues altogether. For the rest of us, we will need to consider it - whether a little or a lot.
I think about stuff too much too.