If you're just training, then it's generally fine if you have the time and money to attend multiple gyms.
If you're actively competing and you're training at multiple gyms which have fight teams and are trying to establish their cred for producing successful fighters, then it might still be fine or it might be potentially politically problematic depending on the relations between the different gyms. In some cases a coach might not give you as much dedicated attention if he feels like you aren't going to be representing his gym in your fights. In some cases (if there is bad blood between the gyms), they might not even allow you to cross over.
In my immediate area, most of the MMA gyms are on good terms with each other and I know several fighters who will drop in to train at multiple gyms. On the other hand I know at least a couple of gyms where the owners strongly dislike each other and if you were to train at both you would at the least not get the best quality attention and coaching for your fight prep.
In my area, I know of a few gyms that, while teach everything, have their own specializations. So some people go to gym a for BJJ, gym B for striking, and sometimes a gym C for standup grappling. Not sure if that's the norm or not, but the ones who train more seriously that I know around here have their specific gyms for each type.
Personally, I can't afford that. Nor do I have the time to go all over the island trying out/going to different gyms.