There really is no way to keep opinion out of an answer to this question, so... 5 man dungeons would be possible with an all druid or all paladin group, but it would be very difficult to be successful in a 10-25 man raiding environment.
Most encounters are designed with certain class abilities in mind. Ex. Heroism/Bloodlust (From a Shaman) or Time Warp (From a Mage). I am not saying that an encounter would be impossible without those abilities, but they become easier with them.
Also, most raid fights require a healthy mix of ranged and melee dps. With this in mind, Druids can be both melee and ranged dps but Paladins are stuck with only melee.
As far as PvP is concerned, it depends on what you are doing. 10 man rated battlegrounds? You would probably not fair very well against a team with a mix of classes because of what different classes bring to the table. Paladins have some major drawbacks in PvP situations, they can be kited way too easily and they really have no true CC (Crowd Control) for peeling and assisting teammates.
Back when Cataclysm first came out there were a few five man arena teams with four Retribution Paladins and one Restoration Shaman. Their tactic at the beginning of the match? The Shaman would use Bloodlust/Heroism and the Paladins would use Avenging Wrath with Hammer of Wrath and just nuke down one or more enemies before they lost one. This group was broken when Bloodlust/Heroism/Time Warp were removed from arenas.
TL:DR - Is it possible to have only all Druids or all Paladins in a group or raid? Yes it is.
Will it be as easy to be successful at what you are trying to accomplish? No it wont.
I am a strong believer in "Bring the player, not the class" however, and what I say is, do whatever makes you happy and helps you enjoy your time playing World of Warcraft.