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.
If you want to be effective in skills and combat, I think you could go straight bard. However, I think it could be extremely effective to take a 1-level dip in Warlock with the Hexblade Patron (in Xanathar's Guide to Everything), and do the rest of your levels in Bard. I would take the dip in your first or second level, but then I would take it straight to Bard 6 by 7th level for Magical Secrets before considering any other dips.
The 1st level of Hexblade will give you:
- 2 Warlock Cantrips
- I recommend Eldritch Blast + 1 other
- If you can use multiple books, you should consider 1 of the cantrips in Sword Coast Adventurer Guide - I would go Booming Blade for flavor (note that Bards cannot get these by RAW, but they are on the Warlock spell list)
- 2 1st level Warlock spells that you can use your Warlock slots for (these do not stack with Bard slots or spellcasting) or your Bard slots
- I recommend Hex + 1 other
- Hex Warrior
- Proficiency with Medium Armor, Shields, and Martial Weapons
- Pick a melee 1-handed melee weapon and use Charisma to attack!
- Now you can max Charisma, shoot for a 14 in Dexterity with medium armor and a shield and pull off a 17 or 18 AC and put whatever other stats in Constitution for HP or Wisdom/Intelligence to improve skills.
This will make you effective in combat. You already get a lot of skills as a Lore Bard:
- 3 skills at Bard 1
- 2 skills from background
- 3 skills at Bard 3
- possibly 1 or 2 from Race
- expertise in 2 at Bard 3
- expertise in 2 more at Bard 10
This will help you be about as effective in combat as a Valor Bard while still getting the skills of a Lore Bard.
Best Answer
WotC includes all (publicly available) rules in their playtest packets, so if it's not in the playtest packet then there are no rules for it at the moment.
As of Feb 28, 2013 no rules for multi-classing in DnD Next have been released.
Update: Here is a summary of the multiclassing rules in the players handbook.
Multiclassing is similar to 3rd edition in that each time you gain a level you choose what class to gain a level in. Unlike 3rd edition, classes now have ability score requirements to multi-class into them, either 13+ in one ability score or two. You need the ability score of both your current class and your new one.