(An addendum to the answers by Thomas and Poncho:)
One disadvantage of IBE (or advantage, depending on the point of view) is that the central authority knows (or can generate) all private keys, i.e. it allows a global key escrow.
For example, if an email address changes the owner (i.e. there is a new person in a company which now should read the mails addressed to an address), the central agency can give this new person the same private key as the previous one had. (Though there is no way to make sure the previous owner has to forget his key.)
A government could force the central authority either to give the central private key, or selected private keys of individual users, enabling decryption of those messages. This is nice for an intelligence agency which wants to read our mail, but less nice for the users who want their mails to be secret.
If the central private key is stolen, all previously sent messages encrypted using the corresponding public key as a component are now decrypt-able (for anyone who captured them before), not just future ones are affected (which could be solved by key revocation).
Thus, this requires even more trust to the central authority than to the certification authorities in certificate-based systems: not only that the authority (or its current private key) was never compromised in the past (thus creating wrong certificates), but also that it never will be compromised in the future.
An analogue to having multiple certificates (by different certification agencies) for one key would be to use multiple (more or less) trusted central (or now actually decentralized) authorities: split a message in parts such that all ones are needed, and encrypt each one for a different authority's public key (and the receiver's identity, of course). The receiver than would have to obtain his private keys from each authority, decrypt all the parts and combine them again to obtain the message. To read a sent message, an attacker now would have to compromise all the authorities, not just one.
(I'm not sure if this was proposed somewhere already, I just thought about it.) Of course, each authority has to check the receiver's identity before handing them the key, which might make this inpractical in practice. One could use some k-out-of-n secret sharing scheme to make not all authorities needed.
Edit: I just found something similar in the last chapter of Advances in Elliptic Curve Cryptography (page 244): Here Alice (with identity $Q_A$) gets secret keys $[s_1]Q_A$ and $[s_2]Q_A$, where $s_i$ are the private keys of two trusted agencies with public keys $Q_1 = [s_1]P$, $Q_2 = [s_2]P$. Bob then creates $$([t]P, M \oplus H_2(\hat e(Q_A, Q_1 + Q_2)^t)),$$
where $M$ is the message, $t$ a random number, $H_2$ a hash function, $\hat e$ a modified pairing. For decryption of a message $(U,V)$, Alice uses the sum $S_A = [s_1]Q_A + [s_2]Q_A = [s_1 + s_2]Q_A$ of her private keys, by calculating $$ M' = V \oplus H_2(\hat e(S_A, U)).$$
This uses the basic scheme of Boneh and Franklin, published in Identity based encryption from the Weil pairing, 2003.
Of course, this is easily generalizable to any number of trusted agencies, who then must collaborate in order to escrow Alice's key $S_A$.