  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 nonprojective measurement is used to generate the randomness locally. We
then prove that the maximum amount of randomness that can be generated using
nonprojective 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 nonprojective qubit measurements with four
outcomes can only be self-tested to a limited degree in a Bell experiment.
