You definitely should be able to, if the ticket was booked and paid for in one motion and even more so if the participating airlines have code shares or belong to the same frequent flyer family.
It sometimes may be more difficult if there is long layover between any of the flights.
Just the other day, I flew from NYC to London, where I planned to spend 12 hrs, and then fly to my hometown elsewhere in Europe. Upon checking in at JFK, I only received the boarding pass for the first leg of the trip, even though the luggage was checked in to the final destination. I was told to go to the transfer desk at Heathrow to obtain the 2nd leg boarding pass. I did not like this as it would require me to shorten my time in London and return to LHR earlier than I wanted. (This airline does not have mobile check in).
However, as I was waking to my first plane at JFK, I walked by a gate where another plane from the 2nd airline was about to start boarding. Even though I was still in NYC, I walked up to the gate agent and very politely asked him to check me in for the flight from LHR to my hometown. He did it on the spot and handed me my other boarding pass. Therefore, I discovered I nice work-around.