Yes, we (adults) dream more than once every night.
The University of Californa has a Dream Research team.
Their FAQ explains:
How often do we dream, and when?
Most people over the age of 10 dream at least 4 to 6 times per night during a stage of sleep called REM (for Rapid Eye Movements, a distinguishing characteristic of this stage of sleep). During REM periods our brains become as active as they are during waking, although not all parts of the brain are reactivated (the parts of the brain that are reactivated in REM are discussed in Chapter 1 of Domhoff's The Scientific Study of Dreams (2003)). REM periods vary in length from 5 to 10 minutes for the first REM period of the night to as long as 30-34 minutes later in the night. It thus seems likely that dreams can be a half hour or more in length.
There is also evidence that we can dream in non-REM sleep in the hour or two before waking up, when the brain has become more activated than it was earlier in the night. That's why we said that we dream "at least" 4 to 6 times per night.
They go on to qualify that there are also dreams while awake, REM periods without dreams, and evidence that children under 10 dream less frequently.