I had mine before 37 weeks on my own, water just broke at 35 and 36 weeks. Both were healthy babies=)
I've had 3 overdue, 1 induced, the other 2 I did have sweeps to get going, my latest one was at 40+6.
Absolutely! Most babies are born between 39-41 weeks but some babies come earlier but still full term on their own. My first baby came at 40wks to the day, on her own. My second baby was induced at 39w4d but I was already 5cm so I would have gone any minute anyway. My third labor started right about 39 weeks by my calculations although my ob swore he was only 37-38 weeks. A day and a half later I gave birth to an 8 lbs boy so I think I was right ;-). So technically all my babies came 39-40wks. I know how hard it is to get to 37weeks and feel done but hang in there! Your babe will be here before you know it!