Noise will become the one that will bite you.
I have done 14/16 bit conversion, but you need to get very fussy about thermal noise, linearity and offsets, and due to sampling accuracy your sampling rate has to go down. There are some low convert rate devices out there at 20 bits I believe, but they are in KHz convert rates.
After you get around the problems with thermal/flicker noise, you can do some digital averaging to reduce the noise some more, but again, your conversion rates have gone down some more.
The prior comments by others are very valid also - some can be solved (the non-linear capacitor can be corrected for with digital curve compensation for example) but there are a large number of subtle things that become an issue.
An antenna, with a 28 bit converter connected to it, converting at 6GHz is not going to happen in the forseeable future.
Never say never!