How can you tell if it's really fat what hiding under your eye lids?

There  is a simple clinical test we use to determine what the bulge is. Take your cell phone and record a "selfie video."  While recording, close your eyes and gently press on your eyeballs.  If you have bulging fat under your eyes, the fat will bulge more when you apply pressure to your eyeballs. Press, release, press, release.  The video will tell you what your problem is.

Fat below the eyelids is contained in two compartments in the upper eyelid, and three compartments in the lower eyelid. After ruling out a history of allergies, or medical conditions that can lead to swelling around the eye then it's possible to proceed with cosmetic eyelid surgery.Excess fat can be removed during a blepharoplasty procedure. In the upper eyelid the fat is removed after excising the excess upper eyelid skin. In the lower eyelid the fat is removed by making an incision inside the lower eyelid. This allows access to the three fat compartments as needed. Lower eyelid skin can be removed if needed by performing a pinch blepharoplasty. This uses an incision directly under the lower lid lash line to remove excess skin. Overall, blepharoplasty incisions heal very well and it's very difficult to determine that someone has had a surgical procedure the patients are left looking more youthful and refreshed.

There are 3 small compartments of fat in each of the lower lids that become herniated over time and create the "puffiness look" or "tired look". A trans-conjunctival lower blepharoplasty  with the incisions placed on the inside of the eyelid can remove the bags which are creating the puffiness. For many examples, please see the link and the video below