When should we not draw a Lewis dot structure?
Short answer: no, there's nothing preventing you from drawing arbitrarily complex molecules according to the conventions of Lewis dot structures. They are not special, and the rules for drawing them are settled by convention and have evolved since they were first developed in Lewis's original paper. They're just one type of diagrammatic rendering of molecular structure which (probably due to their simplicity and explicitness) has caught on in pedagogy. They don't see much use beyond that. Read on for further details.

It's worthwhile to precisely define our terms here. The IUPAC Gold Book provides the following definition for a central atom:


  The atom in a coordination entity which binds other atoms or groups of atoms (ligands) to itself, thereby occupying a central position in the coordination entity.


By that definition, the term is reserved almost exclusively for coordination complexes with a distinct metal atom at the center and ligands bound to it (with some exceptions for purely molecular Lewis adducts, e.g., amine oxides), where the electrons of the bond(s) originate on the ligands, rather than each atom donating an equal number of electrons in the formation of the bond(s). For non-coordinate covalent bonds, if the bonds were to be split homolytically (i.e., such that an equal number of electrons are distributed to each atom), the result would be separate, neutral atoms. Were coordinate covalent bonds to be split this way, that would not be the result.

Obviously, that definition is highly limited, and it's not the prevailing definition in my experience. The other usage of the term is essentially to refer to any non-terminal atom (i.e., one that has two or more bonds to separate atoms). There's nothing inherent to Lewis structures that prevents them being used to represent molecules containing multiple such central atoms. They're simply one representation of bonding. In practice, they don't see much use, chiefly because they're substantially less economical than condensed structural formulas or the skeletal bond-line structures (introduced in their prototypical form by Kekule), and because pure Lewis dot structures, as they were originally conceived, are exceedingly poor at representing stereochemistry.