This is not an "accidental" se. The verb here is romperse, which is pronominal, i.e. it takes a pseudo-reflexive pronoun that refers back to the subject. It's like the verb "to break" in "The window broke" (as opposed to "I broke the window"). So
Se rompió tu ordenador.
means "Your computer broke down." It might be accidental, and it certainly takes away all (hypotethical) responsibility from the speaker, but it's not a particular trick with the verb.
Now if you add a dative pronoun referring back to the speaker, then it means the speaker accepts the computer broke down under his or her watch, so to speak:
Se me rompió tu ordenador.
"Your computer broke down on me."
You don't need the extra dative pronoun to show it was an accident; that job is already being done by the verb itself. If you say “Se rompió” you already mean "It broke", as if by itself.
This dative pronoun need not be the first person singular. It can be any other pronoun: me, te, le, nos, os, les. It can also be a full noun phrase (but the pronoun must be repeated in that case):
A mí se me rompió tu ordenador.
A nuestros padres se les rompió tu ordenador.
Also, of course, the subject of the sentence can be a noun with a possessive (mi ordenador, tu ordenador) referring to any person, or a noun phrase without a possessive (el ordenador).