How much information does a frequency response plot really give us?

I recently had an opportunity to try out a few speakers with considerably different measurements and design philosophies.
On the one hand, I spent a good few hours trying out some of my uncle's Audio Note speakers, which typically measure very tight, +/- 1.5 dB if memory serves.
Then I stumbled into some Focus Audio FS-688 and 788 speakers that cost as much as the AN K's and AN E's, but measure +/- 3 dB's and get even sloppier in their low frequencies.

As much as I think of the Audio Notes, I was really wowed by the Focus Audio speakers.
Intuitively, one might expect the better measuring speakers to sound better, but I'm sure we've all heard speakers that measure well and sound bad, and vice-versa.

This got me to thinking about what's really being captured in these measurements. Obviously the fundamental tones are being measured, but are the harmonics and overtones being accounted for too? What about the impact of attack and decay? I mean, we could have a singer, a tuba, and a guitar making sound at 300 Hz, but the all sound different because of the harmonics or timbre. Does a frequency response plot account for this? And if so, to what degree?

Could we infer from a plot that a speaker with a good response should output the harmonics of an instrument within it's limitations (ie: +/- 3dB or whatever) as well as the fundamental frequency? (provided of course that the recording captured this well)

Thoughts?