From the perspective of a compressor packager, I can say that the pressure in which a unit needs to be blown down to in order to allow restart varies based upon each different compressor package design. The limiting factor is generally the bypass/recycle line size, as they are commonly the smallest line in the process system when recirculating gas. Reason is that just as soon as the compressor begins to rotate, it begins to move gas. The amount of gas it moves is dependent on the pressure that is in the process system; the higher pressure, the higher the flow. As the gas begins to move through the process piping, friction of the gas causes a pressure drop. Another way of putting it is that discharge pressure builds, which in turn increases the power required to turn the compressor. If the power required exceeds what the driver can generate at the low starting speeds, the driver will stall and never achieve the minimum speed.
Bottom line is that figuring out the maximum allowable startup pressure is most commonly a trial and error process. Some calculations can be made to predict the pressure, but most of the time, either the engineering know how to do so, or a knowledge of the required detail of the package does not exist, thus one must resort to trial and error.
Will