We investigate how much randomness can be extracted from a generic
partially-entangled pure state of two qubits in a device-independent setting,
where a Bell test is used to certify the correct functioning of the apparatus.
For any such state, we first show that two bits of randomness are always
attainable both if projective measurements are used to generate the randomness
globally or if a non-projective measurement is used to generate the randomness
locally. We then prove that the maximum amount of randomness that can be
generated using non-projective measurements globally is restricted to between
approximately 3.58 and 3.96 bits. The upper limit rules out that a bound of
four bits potentially obtainable with extremal qubit measurements can be
attained. We point out this is a consequence of the fact that non-projective
qubit measurements with four outcomes can only be self-tested to a limited
degree in a Bell experiment
