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It depends on the depth of the BI, agency involved, priority of investigation (i.e. is OT authorized for the investigator?), investigator workload, etc. They can definitely be done in less, especially for local PD jobs.

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It would seem this way, but it is quite common. I know people that have been hired in that time period—you just need to hit a hiring cycle at the exact right time.

Remember, a police officer can be fired quite easily for the 1st 1-2 years of employment (depending on the agency). So, a lot of departments with hiring crunches take people on before the BI is complete and put them in the academy. As the BI results come in, they fire those deemed unacceptable. 

NYPD is famous for doing this. I have heard plenty of stories from friends who work there about academy classmates being hauled out of class and fired (and in some cases even arrested). 

Not a great practice, I know. Still, it appears to be routine in many large agencies where they need warm bodies on the streets.