Q:

Logic behind "at most"

Does the term 'at most' mean there is at least one?

A:

No because 

"at most one"

means 

one or less.

Edit
The free dictionary points out that at most can be replaced by "not more than".  
In which case 

at most one can be equated to not more than one, 

whereas instead 

at least one means not less than one. 

