Yes, I was found to suffer sleep apnea (temporary absence or cessation of breathing. ). I did not know why I was always tired in the morning everyday, yet had slept. Turns out that I never reached a healthy sleep because I was waking up to take a breath...but didn't know it.
I now sleep with my cpap machine (constant positive air pressure) and wake up refreshed. It took some time to get used to the cpap, so if you get one, don't give up and USE it.
Not sure if this is your problem. I was never given anything for the restless leg syndrome, so I still move a lot while sleeping, but at least I'm sleeping. I typically can keep the mask on for 6 hrs or so, I usually wake in the morning with it off, so I guess I take it off sometime during the night, but the lovely sleep I do get is so restful, I'm ready to go the next day and no longer feel tired. (Added bonus, I no longer snore. Hubby appreciates that!)
They may find something else is wrong with your sleep, but the actual testing in a sleep lab is a piece of cake. The cost was $2,800 a night, but it was worth it to me.
After you have your sleep test, let me know what they find. If you get a cpap, PM me. I have more advice on that if you need it.

Good luck, Rose!