This happens in spectra.  By [a theorem of Lin](http://www.ams.org/journals/proc/1976-056-01/S0002-9939-1976-0402738-5/), there are no maps from the Eilenberg-MacLane spectra $H\mathbb{F}_p$ to finite spectra; the proof is an Adams spectral sequence computation using the fact that the Steenrod algebra is self-injective.

For a more algebraic example, let $A$ be ring containing an infinite regular sequence $(x_0,x_1,\dots)$ and let $M=A/(x_0,x_1,\dots)$.  We can resolve $M$ by an infinite Koszul complex and compute that $\operatorname{Ext}^*(M,A)=0$.  It follows that in the derived category of $A$, there are no maps from $M$ to compact objects.

As for getting some kind of control on these objects, I don't really know much, but I know Luke Wolcott has thought a lot about pathology in derived categories of non-Noetherian rings.  You might try taking a look at his work and seeing if you can find anything useful.