We're not looking for evidence to support Berry's hypothesis, we're looking for evidence to disprove it. It's what family historians ought to do (but often don't).I’m confused now as to what we’re supposed to be doing. It’s very hard to prove or disprove anything without some knowledge of the family - otherwise, particularly with a surname like Adams, we’re somewhat floundering in the dark. How are we supposed to guess if the 1841 and 1861 census entries found support the theory about the family or explode it? I know that too much information could prejudice our searching, but too little doesn’t help either.
If this is a different family it's unlikely this is the only census on which they appear. It's the head of household who doesn't fit into Berry's family - neither his name nor his birthplace is correct. but the others are a pretty good fit, assuming that the birthplaces in the census are interpreted as Leicestershire. The mother is too young to have married in 1833 but the mother in Berry's tree knocked some years off in other censuses, so that doesn't rule her out.
But you need to forget about Berry's family - we're not looking for them! It's the head of household we really need to find.....