The Tesla factory started life as a GM factory, then it became a joint plant run by GM and Toyota called NUMMI. When GM got into financial trouble, they pulled out of NUMMI and soon after Toyota started looking for someone to take the plant off their hands. Right at the time Tesla was looking for a factory.
In the heyday of NUMMI, the plant produced 500,000 cars a year. I hasn't produced as many as 100,000 a year since Tesla bought it. In other words Tesla has a lot of under utilized space. The Model 3 production lines (2 of them to start) are separate from the S and X production lines. So there are no bottlenecks at the factory. However, there is the possibility that once Model 3 deliveries begin that might impact the capabilities of the delivery teams around the country until they get their staff spun up to the new larger volume of traffic.
Tesla has talked about being ahead of the curve on infrastructure and they might succeed this time, but historically they have tended to lag behind production. There was a bad bottleneck at service centers last year and it wasn't until December they started hiring new staff and got cars moving through. They are doing much better now, but they need to step up the game by close to an order of magnitude to support the Model 3.
I expect the Model 3 to have fewer teething problems than the Model S or X had, it's a simpler car than either of those and Tesla has learned a lot of lessons since designing those cars, but all new cars have some problems and they will need some service center support, but they are also talking about expanding the ranger service again.
In any case, production at the factory will only be minimally impacted by Model 3 production ramp up. Any bottlenecks, if they occur, will likely by somewhere else in the pipeline.