I've been having a discussion with my players recently about the game we're running, and am having trouble answering a question that was posed to me.
Why would you ever NOT multi-class in D&D5e, assuming the option to was available to you?
For instance, we have a 3rd-level Barbarian who has started multi-classing in to Druid. He's not going to be getting any more significant Barbarian abilities until 9th-level, so why not gain the ability to cast healing spells on himself, or shapeshift in to a bear whilst raging?
Or a 3rd-level Warlock; what are the downsides to multi-classing in to Rogue (Assassin), so that you can cast and see through magical darkness whilst using the assassinate ability?
The PHB says:
you'll sacrifice some focus in exchange for versatility
but unless you're very strict character roleplayers, I can't see that much focus really being sacrificed. It seems like the phrase "A jack of all trades, a master of none" doesn't seem to apply in this instance; you really can become "a master of all trades".
A thorough answer to this would be appreciated, else I can see my group becoming barbarian-druid-warlock-paladins sooner rather than later.
Best Answer
There are two main downsides to multiclassing, though depending on your play style and group you may never run into the lesser of the two issues.
Falling behind single-class characters
The primary issue, and one that is more likely to impact any game that makes it past 5 levels or so, is that classes scale in power generally at the same time. Casters get level 3 spells (Fireball, et al) at 5th level, martial characters get Extra Attack at 5th level, and Rogue and Bard get Uncanny Dodge/Font of Inspiration.
If you're a wizard and you decide to multiclass into rogue before or at your fifth character level, you will forever be a level behind in your spell progression, and since Challenge Ratings take into account your character level and expect that a level 5 character will have either Extra Attacks or fireball (or other equally powerful spells), your character may have a hard time keeping up damage-wise in combat situations, because you're only a character with level 4 wizard capabilities and level 1 rogue capabilities, neither of which are very impressive compared to what you get at 5th level with either. The only thing you do gain at that point is an increase in your proficiency from +2 to +3, which isn't nearly enough to make up for the initial delay in spell/class feature progression.
Now, it is true that this impacts spellcasters significantly more than it does martial characters, but it does affect your wizard's ability to be the best wizard he can be.
A note on ability score improvements
It is also important to note here that unlike in previous editions, 5th edition D&D ability score improvements are granted by class level, not character level. This means that multiclassing into more than one class can and likely will result in suboptimal ability scores at higher character levels, but when advancing through the tiers of gameplay this effect can be more substantial.
More on ability scores
The multiclass rules (if you choose to use them) require secondary (and tertiary) classes to meet certain ability score requirements. If those requirements are ignored, it is likely that those class levels will be "wasted" as the character will display subpar performance in that class due to their inability to keep up with characters of a similar level. This is less of a problem in 5e due to bounded accuracy, but is still probably enough of a deterrent to multiclassing into more than two classes.
No capstone abilities
The second real drawback to multiclassing, which I mentioned may not affect you based on your group and play style, is that levels are capped at 20 in 5th edition D&D, meaning that characters who multiclass won't ever get their capstone abilities (and possibly 17th/18th/19th level abilities). Once you reach level 20, you can gain no more class levels and thus you can not gain any more class features. Since most games don't reach this level of play, it's not as much of an issue, but is something to consider, especially if starting a game at high level with the intent to play at level 20.
Examples
Using your example of barbarian-druid-warlock-paladins, that's a character that's going to need at least a 13 in STR, CON, WIS and CHA. Unless your player rolled really well it is unlikely they have a character build that can even support this, especially given that the character will likely miss out on one or more ability score improvement throughout her life.
Imagine this theoretical character: barbarian 4/druid 4/warlock 4/paladin 4. This character has a proficiency bonus of +5, and has received 4 ASI's throughout his career.* However, his class features for any given class only equal that of a level 4 character. He is tier 1 in every class, and has no tier 2 abilities that can help his survivability or damage output in higher tier gameplay. He is in a party of APL 16, with 3 single-class characters and the party is facing a CR 16 monster. Which party member is going to be the least helpful in this situation at a high level of play? Probably the character who has only one attack per turn, no access to anything above a level 2 spell, and no ability to wild shape into anything but a 1/2 CR beast that can't even fly.
* Note that this is on par with a single-classed wizard character; however, a level 16 wizard at this point has access to many spells (up to 8th level) that can enhance his abilities and defenses that the multiclassed character does not have access to.