I know in many cases, specifying which sequences converge is sufficient for specifying which topology is being used. I was wondering in which kinds of scenarios is this necessarily the case, and when it might fail
In a metric (or metrizable) space, the topology is entirely determined by convergence of sequences. This does not hold in an arbitrary topological space, and Mariano has given the canonical counterexample. This is the beginning of more penetrating theories of convergence given by nets and/or filters. For information on this, see e.g.
In particular, Section 2 is devoted to the topic of sequences in topological spaces and gives some information on when sequences are "topologically sufficient".
In particular a topology is determined by specifying which nets converge to which points. This came up as a previous MO question. It is not covered in the notes above, but is well treated in Kelley's General Topology.
If you go there, you can find a few examples too.