Sure -- it is CRITICAL to read and review the definitions of LTD and STD.
In some STD policies disability is broadly defeined as the inability to do your usual work, while for LTD it may be more narrowly defeined as the ability to do any work for which one is suited by training or experience. Thus a brain surgeon could easily qualify for STD if illness made her hands shaky, but not LTD as she could still work as a non-surgeon doctor, giving second opinions, etc.
In addition, most insurance companies give much more scrutiny to claims for LTD benefits than STD as the LTD is far more potentially expensive in any one case. (And very often the decision on STD may as a practical matter be influenced by and/or effectively be made by the employer, as large employers essentially pay the claims.) So even if the definitions of LTD and STD are the same, the amount of evidence the seek is greater.
Line up your evidence carefully, and better yet, consult an attorney NOW to help you do so as there is a lot of money at stake and it's worth a few hundred dollars to get your ducks in order.