After computing in 1899 for the first time the value of what is now named "Planck's constant" (despite the fact that Planck has computed both constants that are now named "Boltzmann's constant" and "Planck's constant"), Planck has immediately realized that this provides an extra value, besides those previously known, which can be used for the definition of a natural unit of measurement.

Nevertheless, Planck did not understand well enough the requirements for a good system of fundamental units of measurement (because he was a theoretician, not an experimentalist; he had computed his constants by a better mathematical treatment of the experimental data provided by Paschen), so he did not find any good way to integrate Planck's constant in a system of fundamental units and he has made the same mistake made by Stoney 25 years before him (after computing the value of the elementary electric charge) and he has chosen the wrong method for defining the unit of mass among two variants previously proposed by Maxwell (the 2 variants were deriving the unit of mass from the mass of some atom or molecule and deriving the unit of mass from the Newtonian constant of gravitation).

All dimensionless systems of fundamental units are worthless in practice (because they cause huge uncertainties in all values of absolute measurements) and they do not have any special theoretical significance (for now; such a significance would appear only if it became possible to compute exactly from theory the values of the 2 constants of the electromagnetic interaction and gravitational interaction, instead of measuring them through experiments; until now nobody had any useful idea for a theory that could do such things).

For the number of independently chosen fundamental units of measurement there exists an optimum value and the systems with either more or fewer fundamental units lead to greater uncertainties in the values of the physical quantities and to superfluous computations in the mathematical models.

The dimensionless systems of units are not simpler, but more complicated, so attempting to eliminate the independently chosen fundamental units is the wrong goal when searching for the best system of units of measurement.

My point is that the values of the so-called "Planck units" have absolutely no physical significance, therefore it is extremely wrong to use them in any reasoning about what is possible or impossible or about anything else.

The "Planck units" are not unique, there also exists a very similar system of "Stoney units", proposed a quarter of century before the "Planck units", where the values of the units are different, and there are also other variants of dimensionless systems of units proposed later. None of them is better than the others and all are bad, the worst defect being that the huge experimental uncertainties from measuring the value of the Newtonian constant of gravitation are moved from that single value into the values of all unrelated physical quantities, so that no absolute value can be known precisely, but only the ratios between quantities of the same kind.

In a useful system of fundamental units, for all units there are "natural" choices, except for one, which is the scale factor of the spatio-temporal units. For this scale factor of space-time, in the current state of knowledge there is no special value that can be distinguished from other arbitrary choices, so it is chosen solely based on the practical ease of building standards of frequency and wave-number that have adequate reproducibility and stability.

The only historical value of the "Planck units" is that they provide a good example of how one should NOT choose a system of units of measurement. The fact that they are still frequently mentioned by some people in any other context than criticizing such a system just demonstrates the very sad state of physics education, where no physics textbook includes an adequate presentation of the foundation of physics, which is the theory of the measurement of physical quantities. One century and a half ago, Maxwell began his treatise on electricity and magnetism with a very good exposition of the state of metrology at that time, but later physics textbooks have become less and less rigorous, instead of improving.