= Solution
For null <four-momentum>, $\{Q^I,\bar Q^J\}=2\delta^{IJ}\sigma\cdot p$ has rank one in its spinor indices. Positivity makes the null combinations act trivially; the remaining $N$ normalized pairs obey $\{a_I,a_J^\dagger\}=\delta_{IJ}$. Acting with distinct lowering operators on a highest-<helicity> state gives the <helicity spectrum of a massless supermultiplet>:
$$
\lambda_k=h-\frac{k}{2},\qquad n_k=\binom Nk\quad(0\leq k\leq N),\qquad
\boxed{\sum_k n_k=2^N.}
$$
A <CPT completion of a supermultiplet> adds the opposite <helicities> if needed. Interpreting the physical spin bounds as $|\lambda|\leq s$, the width $N/2$ fits inside $[-s,s]$ only if $N\leq4s$. A literal one-sided bound on helicity without <CPT> would not give this maximum. The maximal spectra are
$$
\begin{array}{c|rrrrr}
N=4:\ \lambda&1&1/2&0&-1/2&-1\\
\text{multiplicity}&1&4&6&4&1
\end{array}
$$
and
$$
\begin{array}{c|rrrrrrrrr}
N=8:\ \lambda&2&3/2&1&1/2&0&-1/2&-1&-3/2&-2\\
\text{multiplicity}&1&8&28&56&70&56&28&8&1.
\end{array}
$$
Thus \b[the maxima are (N=4) for spin at most one and (N=8) for spin at most two], with 16 and 256 states. <Dimensional reduction> of <ten-dimensional super Yang-Mills theory> on a six-torus gives the first: the vector yields one four-dimensional vector and six <real scalar fields>, and its <Majorana-Weyl spinor> yields four <Weyl spinors>. Reducing <eleven-dimensional supergravity> on a seven-torus gives the second: metric components supply one <graviton>, seven vectors and 28 scalars; the three-form supplies 21 vectors, 35 scalars and seven dualized two-form scalars; the <gravitino> supplies eight <gravitini> and 56 spin-half fields. These total 28 vectors and 70 scalars, with 128 <boson> and 128 <fermion> states.
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