I've never had a begonia tuber get too big, mine seem to max out at about 15cm. I wouldn't throw them away as they just get bigger and blousier each year and I've never noticed any deterioration.
You can also save fibrous begonias. I bought some Burning Embers plug plants last year and in late autumn brought the pot indoors to overwinter and they came back this year:
You can also save fibrous begonias. I bought some Burning Embers plug plants last year and in late autumn brought the pot indoors to overwinter and they came back this year: