I will be blunt. As fas I know, nobody knows a priori for which systems equilibrium statistical mechanics will work or not.
Part of the current effort to determine which systems are fine being described by equilibrium statistical mechanics focuses on various proofs of ergodicity for such systems. For now, they are somewhat limited to either a restrictive definition of ergodicity (Birkhoff's theorem for hard sphere systems) or for a broader definition of ergodicity with constraints on the observables to be ergodic for very short range interaction. I would say that the latter attempt was pioneered by Khinchin in his book of the mathematical foundations of statistical mechanics.
Others take on a very different path and consider as fundamental a form of Bayesian inference to justify the use of such a formalism. This line of work was strongly advocated by E.T. Jaynes and more recently by the likes of R. Balian in his book from microphysics to macrophysics.
The existence of attractors is not to be ruled out from the outset. For instance, one can study the thermodynamics of a table and figure out its thermal expansion coefficient, specific heat capacity, Young modulus, electrical properties and so forth. All of this can be done by using equilibrium statistical mechanics (and the canonical ensemble most of the time). Yet, the centre of mass of the table itself does not wonder about (on Earth at least) with some Boltzmann probability in my lounge. This means that the Boltzmann weight is to be used only for some aspects of the motion of the particles comprising the table but not for all. In that sense, most of the statistical thermodynamics we do in practice has to deal with the state of a system in a particular non-ergodic attractor; and that's fine.
Finally, it also depends what you consider as being given to justify the use of the canonical ensemble. For instance, even when given the rule of equally likely micro states for an isolated system it does not always imply that a system made of two parts (one small and one big) will reduce to a Boltzmann weight for the small sub-system. In particular that does not hold if the particles in each subsystem have long range interactions (like gravitational ones). General theorems on this particular assumption (microcanonical implies canonical) have been derived by using large deviation theory.
There is unfortunately no necessary conditions on which every physicist agree to retrieve equilibrium statistical mechanics. However, recent advances have permitted to show what does not constitute a necessary condition for statistical mechanics to work. I would mention:
Chaos is neither enough nor sufficient to claim that a system will equilibrate according to equilibrium statistical mechanics (contrary to what has been thought and taught for a long time). This has been illustrated in the general result provided by the KAM theorem, by the famous Pasta-Fermi-Ulam problem and by the work on this very topic by Vulpiani for instance.
Large system size. Although there are many results that show that many statistical mechanics results actually hold for large system sizes, there are also instances where this is not necessary. One can mention the proof of the Birkhoff's ergodic theorem for various billiards and also some issues that become even bigger as system size actually increases (like in the Pasta-Fermi-Ulam problem).