Since almost everybody who answered before me seemed so keen on STL containers, I thought it would be useful to compile a list of good reasons not to use them, from actual problems I have encountered myself.
These can be reasonably grouped into three broad categories:
1) Poor efficiency
STL containers typically run slower AND use too much memory for the job. The reason for this can be partly blamed on too generic implementations of the underlying data structures and algorithms, with additional performance costs deriving from all the extra design constrains required by the tons of API requisites that are irrelevant to the task at hand.
Reckless memory use and poor performance go hand in hand, because memory is addressed on the cache by the CPU in lines of 64 bytes, and if you don't use locality of reference to your advantage, you waste cycles AND precious Kb of cache memory.
For instance, std::list requires 24 bytes per element rather than the optimal 4.
https://lemire.me/blog/2016/09/15/the-memory-usage-of-stl-containers-can-be-surprising/
This is because it is implemented by packing two 64-bit pointers, 1 int and 4 bytes of memory padding, rather than doing anything as basic as allocating small amounts of contiguous memory and separately tracking which elements are in use, or using the pointer xor technique to store both iteration directions in one pointer.
https://en.wikipedia.org/wiki/XOR_linked_list
Depending on your program needs, these inefficiencies can and do add up to large performance hits.
2) Limitations / creeping standards
Of course, sometimes the problem is that you need some perfectly common function or slightly different container class that is just not implemented in STL, such as decrease_min() in a priority queue.
A common practice is to then to wrap the container in a class and implement the missing functionality yourself with extra state external to the container and/or multiple calls to container methods, which may emulate the desired behavior, but with a performance much lower and O() complexity higher than a real implementation of the data structure, since there's no way of extending the inner workings of the container. Alternatively you end up mashing up two or more different containers together because you simultaneously need two or more things that are fundamentally incompatible in any one given STL container, such as a minmax heap, a trie (since you need to be able to use agnostic pointers), etc.
These solutions may be ugly and add on top of the other inefficiencies, and yet the way the language is evolving the tendency is to only add new STL methods to match C++'s feature creep and ignore any of the missing core functionality.
3) Concurrency/parallelism
STL containers are not thread-safe, much less concurrent. In the present age of 16-thread consumer CPUs, it's surprising the go-to default container implementation for a modern language still requires you to write mutexes around every memory access like it's 1996. This is, for any non-trivial parallel program, kind of a big deal, because having memory barriers forces threads to serialize their execution, and if these happen with the same frequency as an STL call, you can kiss your parallel performance goodbye.
In short, STL is good as long as you don't care about performance, memory usage, functionality or parallelism.
STL is of course still perfectly fine for the many times you are not bound by any of these concerns and other priorities like readability, portability, maintainability or coding speed take precedence.