"You promised me a new car!"
Translation:Obiecałaś mi nowy samochód!
Not really. Putting a Dative pronoun at the beginning of the sentence is possible, but it needs to be the stressed, emphasized form then: "mnie".
"Mnie obiecałaś nowy samochód!" would be correct, but rare, the emphasis on "mnie" is really strong here, it would only make sense if someone else mentioned what 'you' promised to them.
A: You promised me a new plane, Steve!
B: What? You promised ME only a new car, Steve! Why is she getting a plane?!"
On a separate note, there are sentence which are more likely and more common to start with a Dative pronoun, and truth be told, native speakers really often make the mistake to use unaccented "mi". It happens to me as well. But well, it's a mistake.
A good way of finding out whether using "mi" instead of "mnie" is correct is to substitute English "me" with singular "you". Then you have either unaccented "ci" and accented "tobie". And while Polish people can make the mistake with "mi", I don't think that anyone could do the same with "ci", that will be an obvious error to any native.