A couple thoughts -
1. A white shirt partially in the shade is in different illuminations; outside, for instance, the well lit section is mostly illuminated by light from the sky, a clear blue sky perhaps, while the shaded portion is lit by light reflected from the environment - grass, trees, a painted structure, etc. The WB seeks to neutralize the predominate illumination.
2. Noise. The lower the S/N in the sampled area, the more the WB calculation can be thrown off by noisy pixels. It follows that a well exposed area will provide a better sample. One popular WB target, the WhiBal card, is 70% grey, rather than the traditional 18% card for determining film exposure, for this reason.
3. LR provides tools (attempting) to treat the problem.
A. Since LR 4 it has been possible to alter the size of the WB probe's sample from 5x5 up to (IIRC) 17x17 and to see the sampled area 1:1. A larger sample, by averaging a larger number of pixels, can reduce the influence of the noise.
B. The local adjustment tools can alter WB.
C. Split Toning can, in effect, alter WB in highlights and/or shadows.