A comment by @Oliver_C mentioned a study that found snoring to occur during all stages of sleep. The original link had gone stale over time, but I could recover the study he was referring to:
Hoffstein, V., J. H. Mateika, and S. Mateika. 1991. Snoring and sleep architecture. American Review of Respiratory Disease 143(1). 92–96.
Unfortunately, the article is paywalled, so I can only quote from the abstract (my emphasis):
We found that even the self-confessed nonsnorers snored lightly, with significantly smaller frequency and index than the heavy snorers. Sleep architecture was similar in both groups. Distribution of snoring among the sleep stages differed for light and heavy snorers: light snorers snored uniformly throughout all sleep stages, whereas heavy snorers tended to snore more during slow-wave and REM sleep.
This finding is difficult to explain if the claim that "you cannot snore and dream at the same time" was true. If snoring occurs during every sleep stage, including the REM stage which is strongly associated with dreaming, it seems unlikely that dreaming and snoring are mutually exclusive. Yet, note that based on the abstract, the study doesn't seem to measure explicitly the co-occurrence of active dreaming and snoring, so it presents only indirect evidence against the claim.