The disadvantages of setting Max Degree of Parallelism to 1 is that queries that would have benefited from executing with a parallel plan will not be able to and that will likely reduce performance. In some cases it can make a previously parallelized process run orders of magnitude longer and monopolize a single-CPU at 100% in the process.

Was Max Degree of Parallelism set to 1 on your server or are you exploring the possibility of making a change?

If you're deciding whether to set it to 1 or evaluating whether that was the right decision consider leaving parallelism enabled at the default (i.e. set to 0) but raising the bar for parallel plan to be chosen by changing the Cost Threshold for Parallelism

Also know that with a Max Degree of Parallelism of 1 you can always override the setting for specific queries you know will benefit from executing with a parallel plan using the MAXDOP query hint.