this kind of question really gets my curiosity up. the first thing i thought of is what i call the sitar effect. it's more common in acoustic guitars where the saddle has been sanded down too short and the breakover angle of the string is too shallow. it could also be caused by an enlarged saddle slot or nut slot so i'm wondering how old the guitar is.
could you tell us what kind of guitar you're talking about, what kind of bridge you're using, etc.? also what size strings are you using and have you tried replacing the high e with another string or possibly replacing it with one size larger string? a lot of the time problems like this are solved by a process of elimination.