The concept of CDW induced by a Fermi-surface nesting, orininated from the Peierls idea of the electronic instability in 1D metals, has taken root among physicists dealing with actual charge ordering in low-dimensional materials. The rule of the game is to cut out quasi-2D Fermi contours and shift them around until some parts visually coincide, and if a shift is close to the observed CDW wave vector (and even if it is is not), the material is declared to be another
example of a nesting-derived CDW of the Peierls type.  We argue that only a tiny fraction, if any at all, of the observed
charge ordering phase transitions have the same nature as the Peierls instability, defined as follows:

(a) there is substantial nesting of the FS at the right wave vector, as quanitified by a peak in Im \chi₀(q);

(b) this peak is translated (as in the 1D case) into a peak in Re \chi₀(q) at the same vector;

(c) this peak leads to a divergence in the full electronic susceptibility, making the electronic subsystem to be unstable by itself, without a necessity to move around the ions;

(d) in the high-symmetry phase, all phonons are softened at the CDW vector, not just the soft mode associated with the actual CDW.

Using prototupical CDW materials such as NbSe₂, NbSe₃, CeTe₃, we show that these conditions are hardly ever satisfied, and the CDWs in questions are rather structural phase transitions with an underlying ionic, rather than electronic, mechanism. We also show quantitatively why the original Peierls construction is so fragile that it is very unlikely to apply to any real material.