You have received some very thoughtful answers through which I have learned some things here. In my experience, outside of differences in speaker quality, amp/speaker compatibility issues, and sonic preferences of the listener, the bass response is the most difficult to get exactly right and (for a given speaker) is the parameter most influenced by the room. Therefore, depending on the speaker and the room, I can envision conditions where a quality full-range speaker could indeed limit performance of a system.
I recently improved the sound of my system by switching from near full-range speakers to large stand-mounted speakers (two 9-inch woofers in each sealed box) along with two high-quality powered subwoofers. This set-up provides a more consistent bass roll-off above 40 Hz (due to the sealed box), better overall integration of bass in the room (due to the superior control of the dual powered subs), improved bass definition, and overall lower frequency extension to below 20 Hz. The ancillary benefit is slightly better clarity through the midrange.
I recently improved the sound of my system by switching from near full-range speakers to large stand-mounted speakers (two 9-inch woofers in each sealed box) along with two high-quality powered subwoofers. This set-up provides a more consistent bass roll-off above 40 Hz (due to the sealed box), better overall integration of bass in the room (due to the superior control of the dual powered subs), improved bass definition, and overall lower frequency extension to below 20 Hz. The ancillary benefit is slightly better clarity through the midrange.