Can QM be used to model 2 state systems with more than 4 linearly independent observables?

Suppose I have a system (e.g., a particle) and I have different physical measurement apparatus which can act on it. Each of the measurement apparatus (observables) has 2 distinct labeled outcomes, thus in QM I can represent them by 2-dimensional self-adjoint operators or alternatively 2×2 Hermitian matrices. I know these observables act on the same Hilbert space since I can experimentally determine that these 4 observables don’t commute.

I can experimentally determine if the observables are mutually linearly independent by taking expectation values, e.g., I can check to see if $S_X$ observable is linearly independent from $S_Y$ and $S_Z$, by checking to see if $langle x_+$|$S_X$|$x_+rangle$ can be made from linear combination of $langle x_+$|$S_Y$|$x_+rangle$ and $langle x_+$|$S_Z$|$x_+rangle$ and same for $langle x_-$|$S_X$|$x_-rangle$.

From linear algebra we know, $2times2$ complex hermitian matrix has 4 free parameters. If we remove the global phase parameter (i.e., $S_I$), there are only 3 free parameters left, hence for convenience we usually represent the observables using a basis consisting of the 3 Pauli operators and Identity, i.e., $S_X$, $S_Y$, $S_Z$ and $S_I$. My question is what happens if I experimentally determine a set of more than 4 linearly independent non-commuting observables, it appears from my perspective regular QM cannot model such circumstance.

I want to know where did my logic go wrong or why such scenarios cannot physically occur? If such scenarios could theoretically occur, how can we go about modeling them? I stumbled upon this question because I didn’t understand why 2 level systems have the structure of $text{SU}(2)$ / $text{SO}(3)$ or why the canonical quantum observables have the algebra they do. I know it is physically true and makes sense since spin can be added with regular angular momentum, but physically why can’t 2 level quantum systems have say 5 or more "mutually unbiased bases", is it just restrictions based on the mathematics behind our theories?

There might be a lot of math I might be missing, I am lost on this topic and I would greatly appreciate pointers on what/where to look for relevant materials.

Edit: This comment has a similar idea to mine