Is it true that the grouping of animals into vertebrates and invertebrates is not logical?

There is more logic to the process than illogic.

The only problem comes with the Protochordates and the Urochordates, two groups of primitive animals who have ill-defined physiologies in the noto-chord/vertebra areas. They tend to 'change their minds' about their structure throughout the course of their development.

This should not be viewed as true difficulty, however, as incremental change is what we expect from an evolving system.