[AccessD] OT...sort of...general db question

Hi all...long time since I've asked for any help, but something came up today, which makes me curious...
 
We have a program, which uses SQL as its db host. There are a bazillion tables...an estimate of course...in this thing, and all but one is indexed. The table that is not indexed is the 3rd most populated table, and it is several tens of thousands of records.
 
My question is this: what would be a programmer's reason to omit an index, for a table? 
 
I understand that one cannot speak for this specific program, but I am just looking for a general answer. It has always been drilled into my head, "index, index, INDEX!" So, I'm wondering if there are times that I may NOT want to include an index.
 
Thank you all for your time.
 
John W Clark
 
John W. Clark
Computer Programmer 
Niagara County
Central Data Processing