बजे is always used to refer to the time. You use 'साढ़े' to say 'half past'. For instance, half past five is साढ़े पाँच बजे.

I think the source of your confusion is that Hindi has words for 1.5 and 2.5, डेढ़ and ढाई respectively. So, half past one is called डेढ़ बजे and half past two is called ढाई बजे.