It has been known for almost two decades that the water proton NMR signal of diffusing water molecules in brain white matter undergoes a non-monoexponential decay with increasing diffusion gradient factor b. With the help of numerical simulations and analytical expressions, much effort has been directed to describing the signal decay and to extracting relevant biophysical features of the system under investigation. However, the physical basis of such nonmonoexponential behavior is still not properly understood. The primary difficulty in characterizing this phenomenon is the variation in behavior in the different directions of diffusion measurement. A combined framework that accounts for the diffusion process in all directions requires several parameters. Addition of many such parameters renders a model to be unwieldy and over-complicated, but over-simplifications can be shown to miss crucially relevant information in the data. In this thesis, I have attempted to handle this problem with simple measurements that span a wide range of parameter space. Compared to often-performed measurements that probe diffusion over a time-scale of 50-100 ms with relatively low diffusion weighting, the measurements here have been done for very short diffusion times of 2 ms and also very long diffusion times up to 2 s. The temperature dependence of the diffusion coefficients has also been extensively probed. To avoid problems related to gross tissue heterogeneity, diffusion-weighted MR imaging in vivo was performed with ultra-high resolution. These simple measurements allowed sequential assessment of many possible arguments that could have led to such non-monoexponential decay curves. Finally, it was concluded that the water in the glial processes was the major contributor to the non-exponential decay, giving rise to a \''slow\'' component both along the axonal fibers and transverse to them.