Based on the content of your cleavage cocktail, I personally don't think that any tbutyl group will still remain intact if ever there has been any alkylation taking place. TFA is a very strong acid used in the cleavage of tbutyl group and BOC group. You can consult the review article entitled :'Trifluoroacetic acid: Uses and recent applications in organic synthesis' which is found in the following link: http://www.sciencedirect.com/science/article/pii/S0022113913003163

What was the cysteine (Cys)  residue protected with? t-Butyl groups are more difficult to remove from Cys, than other residues like threonine and serine. I would suggest an alternate "Cleavage cocktail", if you used t-butyl protected Cys, see links.

http://www.tandfonline.com/doi/abs/10.1080/00397910008087375

http://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=3&cad=rja&uact=8&ved=0ahUKEwiego7pjanRAhVL7WMKHeA7D_UQFggtMAI&url=http%3A%2F%2Fonlinelibrary.wiley.com%2Fdoi%2F10.1002%2Fhlca.19790620744%2Fpdf&usg=AFQjCNEDnuYVatO3VhuM86TtbkSWdBt3tw

I have also encountered the same problem, but I personally don't think that cysteine was alkylated with tButyl group. Sometimes there is no any cysteine, additional mass of 56 Da for the peptide have still occured.

Add some EDT to your cleavage cocktail to prevent t-butylation of cysteine

 Additional mass of 56 Da was still occured  when we added 3% EDT to cleavage cooktail.

Yes it can unfortunately. Even with TIPS, H2O and EDT a t-butyl group can stick on there. It helps to use trityl protecting groups for cysteine and if you are really struggling then use S-tbutyl (di thiol) protecting groups and avoid all thiols until after purification (then chuck in some cycteine hydrochloride and de-salt).