I have seen different values mentioned but the 3.3--7 transactions/second appears to pop up a lot. In Section 2 of On Scaling Decentralized Blockchains (A Position Paper) it says the following:
There are two interesting scenarios: The first scenario is when the Bitcoin network is operating at maximum throughput, namely 3.3–7 transactions/sec. This maximum throughput is mainly constrained by Bitcoin’s 1MB maximum block size and the variable transaction size. The lower bound of the maximum throughput is inferred from the current average transaction size, about 500 bytes, while the upper bound is based on an oft-cited estimate from  which corresponds to unusually small (250 byte) transactions.
The maximum block size is 1,000,000 according to the source code. My calculations give me these values for average transactions sizes of 500 bytes for the lower bound and 250 bytes for the upper bound:
lower bound: 1,000,000 / 500 / 600 = 3.3333(3) upper bound: 1,000,000 / 250 / 600 = 6.6666(6) lower bound’: 1024*1024 / 500 / 600 = 3.4952533(3) upper bound': 1024*1024 / 250 / 600 = 6.9905066(6)
According to my calculations, the correct rounded maximum throughput of the Bitcoin network is 3.3--6.7 for the transaction averages given above. I am wondering whether I got something wrong. The authors could have rounded only the upper bound but that doesn't make much sense though.