I believe this is a case of "learn them as they occur", but I'd like to expand a bit on that. Bear with me; the last examples are the best.
Aburrido is not the only instance of this kind of anomalous behavior, but probably the one that most clearly demonstrates it. There are a few more, not exactly like it, where one could potentially argue that an active, instead of passive, form should have been used:
- vestido
- creído
- pagado (de sí)
- mal llevado
- mal hablado
- pesado
- cansado
In English you say you're wearing (active) some clothes, and you can also say that in Spanish; when no specific clothing is mentioned, both English and Spanish resort to passive forms: Estoy vestido = "I'm dressed". But in Spanish the most usual way to express "I'm wearing such and such" is using passive forms: Estoy/Voy vestido con/de... or Tengo/Llevo puesto/a/s.... We can say Estoy vistiendo tal y tal cosa, but it's not common. The fact that one usually dresses oneself (reflexive) must influence the choice of a passive.
Creído, meaning "conceited", is another example. What is believed (creído) by the conceited person is that he's specially good or able; i.e. the passive meaning should point to the things believed, not to the person that does the believing. In Spanish we have creyente for the meaning "believer" in the religious sense, which might explain why it's not available for this other meaning. In English, obviously, "conceited" is passive, but "conceited" can be quickly translated as the equivalent passive "fooled (by one's own abilities or appearance)"; the Spanish creer has a different grammar, and creído in this meaning is therefore also anomalous.
The same logic applies to pagado de sí, mal llevado and mal hablado. Mal llevado is rather clearly anomalous as a passive: one says for example Me llevo mal con mis primos "I don't get along with my cousins", where llevar means "to conduct". The adjective phrase should be mal *llevante or something like that. As with vestido, perhaps the pseudo-reflexive grammar and the fact that one doesn't do it on purpose trigger the shift to passive ("I conduct myself badly" = "I'm badly conducted"). With mal hablado there are no excuses; it should really be mal hablante or mal hablador, since it's a (mostly) intransitive verb and this kind of passive transformation is forbidden by its grammar.
Pesado works a lot like aburrido. Logically the word to refer to something or someone that is a bore should be *pesante or the like. In a phrase like Tu amigo es (un) pesado, the friend is doing the work of being a bore; he's the metaphorical "weight" (active), he's not being weighed (passive, pesado).
The same as pesado can be said of cansado (entry #3 in the DLE). Indeed, pesar, aburrir and cansar are interchangeable (in syntax) and near synonymous in constructions like Me aburre/pesa/cansa (este trabajo). I never use cansado in this active sense (it's not in my dialect), and it sounds as odd to me as aburrido must sound to an English speaker trying to match the pair "bored - boring" to an equivalent pair in Spanish and finding only aburrido.
To summarize, there are a nice handful of more-or-less "illogical" adjectives derived from passive participles which should be active. Spanish has means to create such active adjectives from verbs (for example, for something that tires you, cansador) but in some cases it chooses not to. I think the meaning of the passive suffix -ado, -ido has simply been forgotten in these cases; it works just as a generic derivational suffix.
