I would say it is possible - and you might need to push your GP hard for a referral to ENT.
My ds had it - never slept through- and found it difficult to nap during day - would wake distressed at least every 2 hours at night...was grumpy miserable and stroppy (worse than terrible two's behavioiur) during the day...snoring also a big sign.
He also had enlarged adenoids (tonsils were huge), and was finally referred for an op. to remove tonsils, and adenoids and put in grommets (adenoids were obstructing eustachian tube giving him severe hearing loss due to chronic glue ear) - It was only acted on because he had speech delay (due to the hearing loss)
We pointed out the large tonsils and poor sleep (and also inability to eat certain food, like bread, which balls up in the mouth) to different GPs and got no-where.
The ENT consultant we finally saw said that we could have an apnoea test - the community team come to your house and put a monitor on your child to find out if their oxygen levels drop during the night - they didn't bother testing ds because it was pretty obvious he had it.
After the post-op swelling went down, he started sleeping through at last - and we can no longer hear him breathing/snoring from the bottom of the stairs.