The ultimate lofty goal of cladistics is to create a phylogenetic tree of all life on Earth in which each branch is monophyletic. Any paraphyletic (or worse, polyphyletic heaven forbid!) branch indicates an error, or in other words a misunderstanding of how the creatures on that branch evolved.
Recent advances in genetics and molecular studies have made finally made it possible to essentially directly test relationships between species, resulting in some large-scale (and messy!) revisions of the phylogenetic tree. Families, orders, and other taxa are being unceremoniously lumped together, split up, and moved around at an unprecedented rate in an on-going wide-spread effort that promises to continue far into the foreseeable future.
A typical example is when the shared physical traits of a group of species are are discovered to be derived multiple times from different, often unrelated ancestors. Such errors can often be fixed simply by splitting up the offending polyphyletic group, and hanging the fragments from more appropriate branches. [Need example?]
Another common but more subtle problem is a small, often highly specialized group that turns out to be closely related to another branch. For example, the old milkweed family (Asclepiadaceae) turns out to be very closely related to dogbanes (Apocynaceae). In fact, they are more closely related to some lianas in the Secamonoideae subfamily than to other more celebrated members, such as periwinkle (Vinca). This problem is generally fixed by expanding the definition of the (paraphyletic) Apocynaceae to include the old (but monophyletic!) Asclepiadaceae family. Thus Asclepiadaceae "disappears" and becomes a subfamily of Apocynaceae, where you may now call it if you enjoy pronouncing absurd heptasyllabic names Asclepiadoideae.