It's not most cases. It's entirely contextual. Even with your example, the year is significant because that is over 30 years ago, giving the entire statement historical context rather than leaving the recipient wondering if this was recent events. "It was January 4th..." (7 months ago?) "1994" (oh, a long time ago) "when I last saw Nirvana" (oh, obviously this is a historical event).

Had you said you saw them in 1994, the month and especially day are likely irrelevant unless the story involves the weather, association with an event, or your story has something to do with how close to was to the singer's death. Kurt Cobain died April ("this year?" - someone unfamiliar) 5th (of??) 1994 (oh wow, right by the date you already read in this conversation, providing plenty of context).

It takes all 3 bits of information to make a complete statement. Any single bit can be used alone effectively to convey enough meaning. You could say you saw them in 1994, a few months before his death, and provide plenty of context to convey the meaning. You could say you saw them 1994 (oh wow, how close to his death?) January (3 months prior) 4th (I do not care about the 4th)

Ill die on this other 3rd hill