Yes. As with anything, blockchains have advantages and disadvantages and should be used for appropriate purposes.
Depending on your perspective or usage, the following may be seen as drawbacks:
Some blockchains are 'slow'. Blockchains move (or process information) at a certain speed (and different blockchains move at different speeds). For example, the Bitcoin network processes blocks on average every 10 minutes. It is recommended that one wait for 6 confirmations (ie. blocks) before feeling confident that a transaction will not be 'reversed' due to a 51% attack. 6 confirmations * 10 minutes = 60 minutes. In a point-of-sale situation with a merchant accepting bitcoin (ie. a coffee shop), it's not practical to wait an hour after a customer has purchased to give them their merchandise. (This drawback can be overcome with second-layer solutions like Lightning Network, blockchains with faster confirmation times, or other technologies like masternodes with InstantSend and ChainLocks.)
Blockchains can only handle a certain amount of data/transactions per block. If a blockchain becomes overwhelmed with use, transactions back up and don't get processed in a timely manner, similar to bumper-to-bumper traffic stuck on a freeway. This leads to longer wait times and higher fees as users compete with each other to get their transactions processed sooner. (This happens frequently with both Bitcoin and Ethereum.)
Storage space. As blockchains age, they get bigger and bigger, requiring more and more disk space. At a certain scale, only those with huge amounts of disk space will be able to keep full records of blockchains. This decreases decentralization and potentially invites abuse or attack.
Energy usage. As Proof-of-Work blockchains grow, they require ever greater amounts of energy usage. Depending on the source of that energy, this can be detrimental if the source is causing harm or is at risk of running out (ie. coal plants, fossil fuels).
Privacy (or lack thereof). On public blockchains like Bitcoin, all transactions by all participants are available to be seen on the blockchain. While identities are pseudonymous on Bitcoin, if a pseudonymous identity is ever revealed, every transaction that person has ever made can be tracked. It's like having your bank statements open to the entire world. In cases where privacy is desired or required (not just for illegitimate purposes), this is unacceptable. In this case, blockchains with privacy technology should be used. On the flipside, when private blockchains are used, transactions cannot be traced. This may be seen as a drawback to authorities and governments who wish to track transactions.
One of the drawbacks is also one of blockchain's strengths. Immutability. Theoretically, once something's on a blockchain, it's there forever. One way this may be seen as a drawback is in the case of social networks built on blockchains. With traditional database-driven social networks, posts can be edited or deleted. With blockchain-driven social networks, posts can never be edited or deleted. Even with additional layers built on top of blockchains (like websites or apps) that could block, screen, or filter posts, anyone with a block explorer can look at the original post on the blockchain itself. So, if you ever change your mind or regret something you post on a blockchain-driven social network, you'll be unable to remove it.