Vorgestern hätte ich fast einen schlimmen Autounfall verursacht. - I almost had caused a car accident yesterday (but you did not). This is subjunctive. Vorgestern habe ich (fast) einen schlimmen Autounfall verursacht. I (almost) caused an accident yesterday. That actually happened. There is a grammatical difference, but none really in meaning. But when one says 'haette', you already know that it did not happen, because it is the subjunctive. I implemented DonHolgo's suggestion.

The "hätte" alternative has an implicit "but ..." or "if not ..." and is perhaps a bit more common, but I don't really see a difference in meaning.

I'm not a native German speaker but I did stay in a Holiday Inn last night (stupid joke)... also the parallels to English are strong. You're referring to the past, so just as in your English sentence you used the preterit "had", not the present "have" likewise in German you need to use the preterit "hatte" (without the umlaut) not the present "habe". The form with the umlaut "hätte" is the preterit but of the subjunctive mood (known in German as Konjunktiv 2) and is not the right form here. ENG DEU have habe had hatte <----------- this is the CORRECT form (Preterit Indicative = Präteritum Indikativ) had hätte Hopefully a native speaker will come along and confirm my explanation. ;)