vegeta super saiyan 3

Should vegeta have turned super saiyan 3? Throughout dragonball Z Vegeta has always pushed himself to the limits trying to best Goku resulting in him becoming a super saiyan 1 and 2. In GT he becomes super saiyan 4 and skips out 3. I think because of his obession with becoming more powerful than Goku would have caused him to have reached super saiyan 3 shortly after Goku did