Yes you can, but only when snoring that occurs during the REM stage of sleep.
“Social” or “common” snoring is the type of snoring that most people experience from time to time and is usually not a cause for immediate concern.
With this type of snoring you CANNOT dream when you snore.
The reason why you do not dream during social snoring is that you are in a state of NON REM sleep and dreams ONLY occur during the REM sleep stage.
Snoring due to Sleep Apnea is a much more serious and dangerous type of snoring and a vastly different problem that you should seek medical advice on as it COULD be fatal.
Sleep Apnea snoring occurs during REM sleep, which is the same stage of sleep that dreams occur in.
During REM sleep snoring, you may be dreaming as well.
There is a caveat to the statements above.
Social snoring 99% of the time occurs during non-REM sleep and Sleep Apnea snoring 99% of the time during REM sleep.
It can, therefore, be inferred that REM sleep determines the ability to snore and during each snoring type’s 1% exception state, dreams and snoring can occur, which means dreams could, theoretically, occurs with any type of snoring.
I have had contact from a number of people over the years that suffer from Sleep Apnea and especially Obstructive Sleep Apnea and a many number of them have told me about how they have awoken suddenly from a snoring deep sleep state (verified by bed partners) recalling some form of dream to a greater or lesser extent.
Of course there is nothing scientific about this “hearsay”, but I do think it is nonsense that the scientific community as a whole simple says that you cannot dream when you snore simply because they haven’t carried out verifiable and specific tests and studies to prove it.