There are two reasons why the "edit" link might not be visible to users that don't have full edit privileges on "normal" (not locked) posts:

As stated in the [FAQ](http://meta.stackoverflow.com/questions/76251/how-do-suggested-edits-work):
> Suggested edits are held in a peer review queue of a fixed size. If the queue fills up, no more edit suggestions will be allowed until the queue has some empty space.

If the queue is full, the edit link does not appear for users without full edit privileges.

The other reason is if there already is a pending edit on that post. Users that can review suggested edit see a different edit link, but users without that privilege simply don't have the edit link.

(I don't know whether users that have had too many rejected suggested edits still see the link but can't submit their edits, or simply don't get the link anymore.)