Basically, no.
If you are struggling to produce good images from the 60D, it's probably down to the lenses you are using with it. This sort of high density cropped sensor will show up the limitations in most consumer grade zooms, and will produce better images if you use primes, or professional zooms.
What lenses are you using with it, and what sort of photography are you taking?
There are three advantages to full frame 35mm lenses from a quality perspective:
1 - more light gathering, so better performance in low light.
2 - availability of wider lenses, so ultra-wide is more readily available.
3 - narrower depth of field for a given angle of view, so you can isolate the subject more easily (with the right lens).
For general photography though, it's pretty much of a muchness.
I shoot digital from 1.5x crop through 35mm full frame, to medium format digital, and am happy that they all have a place. My fuji 1.5x crop is great, and i don't miss the sensor being the size it is.