Before answering your question, it is worthwhile to note the difference between individual specific and population averaged models for clustered data, since they are each interpreted differently:
Individual specific models explicitly model the dependence at the individual level (e.g. by using random effects). In these models the regression coefficients are interpreted as: the increase in an individual's average response for a one unit increase in the predictor - this sounds like the type of model you'd ideally fit but cannot.
Population average models (e.g. GEE) average over the individual/group level perturbations to produce an estimate of the population average effect that "adjusts" for the dependence, in some sense. So, the regression coefficients are interpreted as the increase in an population average response for a one unit increase in the predictor - this sounds like the type of model you did fit.
The two interpretations are only equivalent in linear mixed effects models since averaging over the random effects on a transformed scale (e.g. logistic) does not, in general, produce mean 0 perturbations on the original scale of the data. For more discussion of this difference see my answer here.
Now to answer your question -- the regression model you have fit assumes independence between units and can be conceived of as a GEE model with a working independence correlation structure. By not modeling the dependencies at the individual level, you've effectively averaged over the random effects. Therefore, the interpretation you've given is not unsupportable - the coefficients should be interpreted as the population average change in the log-odds of one extra question being answered (which is pretty close to what you've written), not the expected change in an individual's log-odds of one extra question being answered (you'd need a mixed model to estimate that parameter).
Edit: I've only given commentary on interpretation of your point estimates. One should note that if you've just used the $p$-values produced from an ordinary ordinal logistic regression model, then your inference may be biased. You may want to consider using a bootstrap procedure or deriving robust Huber-White sandwich standard errors for your model to get approximately unbiased confidence intervals and $p$-values.