A "bare majority" could be a "majority of one" but could be higher than one in some contexts.
The Democrat candidate beat the Republican with a bare majority of votes - 50.3% to 49.7%.
The actual number of votes is less important than the closeness of the vote.
In the original context given, if the committe or panel had, say, eleven members then a "bare majority" would imply 6 to 5, i.e. a majority of one.
Incidentally, depending on the voting system being used in any particular context, one might obtain a majority or a bare majority with fewer than half the votes.
Candidate A, 22 000 votes.
Candidate B, 21 900 votes.
Candidate C, 18 000 votes.
Abstentions and "spoilt papers", 8 000.
Although candidate A has the most votes, and a bare majority over candidate B, candidate A has got less than half the total votes cast.
Depending on the electoral rules in place, candiate A might still be delared the winner and described as having "a bare majority" because the result is very close and if a few people switched allegiance the overall result could have been different.