My wife has been creating a giant aquarium in our smp server with about 10 other people. We are playing in vanilla, and are not running any mods or plugins except for tracking for griefers.

My wife takes the dolphins into the tank and names them. The tank is rather large and has a walled off air portion on the top. Despite 4 attempts, the dolphins will despawn even when named and with the appropriate amount of air. When the area is unloaded, despite their named status, they will despawn with no death message.

To reproduce our first try, just use a lead to put a dolphin into an appropriately sized (approximately 12x14x12) tank. Name them, and then unload the area by traveling outside the chunks or going into the nether (we did both when testing and it worked both times). 

The 2nd way is to use a spawn egg to spawn the dolphins in and give them a name tag, then repeat the last step of unloading the chunks.

Attached are screenshots of the tank and dolphins.