Date: 201508
Local time of day: 1201-1800
Location: ZZZ.Airport, US
Flight conditions: VMC
Light: Daylight
Aircraft: B737-800
Operator: Air Carrier
Operating under: Part 121
Flight plan: IFR
Mission: Passenger
Flight phase: Takeoff / Launch
ATC / advisory: Tower ZZZ
Reporter organization: Air Carrier
Reporter function: First Officer
Reporter qualification: Flight Crew Air Transport Pilot (ATP)
Human factors: Workload; Fatigue; Time Pressure
Second reporter function: Captain
Anomaly: Deviation / Discrepancy - Procedural FAR; Deviation / Discrepancy - Procedural Published Material / Policy
Detector: Person Flight Crew
When detected: In-flight
Result: General None Reported / Taken
Contributing factors: Weather; Company Policy
Primary problem: Company Policy

Narrative: Originally our day was scheduled at 8 hour 10 minute block time. We diverted to airport ZZZ and eventually made it to ZZZ1. This put our daily estimated block time at 8 hours 59 minutes. During our preflight procedures the Captain and I had numerous discussions with crew scheduling; Dispatch; and the Chief Pilot on Call. We were discussing the fact that someone had changed the scheduled block time from ZZZ1-ZZZ from 1 hour 10 minutes to 59 minutes. The burn time calculated by Dispatch was 44 minutes. This gave us 15 minutes plus the 44 minute burn to taxi out; takeoff; and taxi into the gate in ZZZ in order to stay compliant with the FAR 117 rule of 9 hours of block time.We questioned how Dispatch came up with that number with the amount of weather in the ZZZ area. We were told that as long as we taxied out in 12 minutes and took off we would be legal. We taxied out to the runway. Upon reaching the runway we were at 10 minutes since block out. We took off thinking we were legally at or under the 12 minutes since block out. After departing; we noticed that the ACARS marked our OFF time at 13 minutes after block out. When we got closer to ZZZ we heard that aircraft just got out of holding due to weather in the ZZZ area. Although I was not on the phone with Dispatch; I don't believe that holding was ever mentioned to the Captain by the Dispatcher. Had it been the burn and therefore our projected block time should have been increased making us illegal to depart. We ended up blocking in at the gate in ZZZ at 9:06 block time for the day.I don't think it is right that Dispatch and/or Crew Scheduling can adjust the projected block times. Originally ZZZ1-ZZZ was scheduled at 1 hour 10 minutes. It was adjusted to a lesser amount (even with lots of weather and come to find out enroute holding in the ZZZ area) to one minute below the 9 hour limit.


Second narrative: Flight was dispatched knowing the pilots were bumping against the nine hour flight time limitation. Someone in operations came up with a number that would allow us to complete the flight at 8 hours 59 minutes. This required reducing the block time well below the normal for ZZZ1 to ZZZ. After both the First Officer and I had numerous discussions with schedulers; Dispatchers; and a Chief Pilot on Call; it was determined that we could depart legally (based on the flawed information we were given) but we would need to be wheels up within 12 minutes of push.We were informed that ATC and operations at ZZZ1 were 'in the loop' and would expedite us. We were told that if we would fly the leg; there would be no problem getting us to the runway and on our way. None of this happened. Someone told us a nice tale to get the flight moving. The problems began with the push. We were cleared to push and parking brake was released (starting the clock.) Around this time; we were told to hold the push until the aircraft at the adjacent gate could move. After about two minutes; the push commenced and we got engines running. Taxi was then delayed waiting for another aircraft to start moving ahead of us on the taxiway. We eventually made our way to the threshold with two or three minutes to spare before becoming illegal. We were given line up and wait instructions as the clock reached XA:18.We were subsequently issued takeoff instructions and both of us checked the clock prior to initiating the takeoff roll; XA:18. Since I never used a stop watch to time a takeoff; it didn't occur to me that the computer would record a time other than XA:18. We found out later that the recorded wheels up time was XA:19. Had I not been nearing a fatigued state; I might have reverse engineered the time requirement and refused the takeoff clearance. I think that the Latest Wheels Up (LWU) number should have a two minute buffer denoted for starting the takeoff roll.Clearly ZZZ1 Operations knew we had time constraints. Unfortunately; there was zero coordination on the ramp. Otherwise the push at the adjacent gate would have been held until we departed and we wouldn't be having this conversation. Further; the Chief Pilot on call provided zero guidance. He deferred to the Crew Scheduler. Also; ATC was never in the loop. They gave us normal ground handling getting to the runway and sequencing us for takeoff.The Crew can no longer trust that there is any sort of team effort taking place. We are on our own. From now own; I will consider LWU a suggestion and won't go onto the runway without a two minute plus cushion. Also; once a divert takes place; it should be SOP to query the crew about their fatigue status. Any experienced airline pilot knows the toll a diversion takes. However; having done this all before; we press on. This is why someone outside of the cockpit should be making an effort to coordinate the status and condition of the crew in light of all surrounding factors (big picture). Instead; the only thing that seems to matter is pushing the flight.


Synopsis: The 737 crew violated FAR 117 due to a divertion followed by a taxi/takeoff delay.