On computing the number of independent values from a signal's bandwidth? Can anyone help me understand where the value of $2$ comes from in the following statement? 

A time function t whose frequency response is white out to a bandwidth B has 
about 2 · B · T independent values when measured for the time interval T.


This is in the context of a serial stream of digital (e.g. binary) data. The function t represents this data stream.
It comes from the Nyquist rate.  The Nyquist sample rate for a signal with bandwidth $B$ is $2B$.  In other words, that is the lowest sample rate that will contain all of the information in the signal without distorting it (i.e. without losing information).

The question implicitly assumes that the sample rate is $2B$, and thus that in time $T$ you will get $2B*T$ independent samples.  If the sample rate was greater than $2B$, then the signal would not be "white" anymore and the samples would have some correlation.