Many dependent variables, few samples: is this an example of "large $p$, small $n$" problem? "Large $p$, small $n$" typically refers to "many independent variables, few samples".
In my case, I have $1$ independent variable, $300$ dependent variables, and $n < 20$ samples.
Thus, my case is not the typical "large $p$, small $n$" scenario (even though I do have many regression coefficients to estimate -- it's just that they are not due to the independent variables, but due to the number of dependent variables).
How is my situation classified?  Does it fit into the "large $p$, small $n$" scenario, albeit I have actually less independent variables than samples?

Note: I would like to do a multivariate regression or MANOVA on my data, but have a "multicollinearity" problem.  But it is not really multicollinearity because I only have one independent variable.  What's going on here?
 A: There are several combinations on the size of $n$ and $p$: small $n$ - large $p$, small $p$ - large $n$, large $n$ - large $p$ ... See Johnstone & Titterington, 2009, Statistical challenges of high-dimensional data for an overview.
In your case, it appears that you have a small $p=1$ and relatively small $n$, with high dimension of the dependent variable. It is likely that your independent variable may not contain enough information to properly model $300$ responses.
The justification of this claim is as follows. If you use a GLM for your data, then you have $20$ samples to estimate $300+$ parameters in the covariance matrix of the errors. This may induce over-fitting, and the precision of the estimators will be unnecessarily vague (in the sense that confidence intervals for these parameters might be too wide) and inaccurate (far from the true value). However, if you restrict the structure of the covariance matrix, then it may be possible to estimate the parameters more accurately (How to restrict the structure of the covariance matrix? That's a big question which depends on the context). Moreover, the fewer covariates you use, the more "responsibility" the residual errors carry to explain the unobserved variability. This may, for instance, inflate the variances or induce the need for more flexible distributions than normal for modelling the residual errors.

Additional references of possible interest:


*

*West, 2003, Bayesian Factor Regression Models in the “Large p, Small n” Paradigm


*CV question: Summary of "Large p, Small n" results
A: [As I read it, the question is primarily about terminology, and @East's answer (good as it is) does not explicitly address it.]
Sometimes the distinction between dependent and independent variables is not so clear. As you are referring to MANOVA, you probably have $300$ variables measured for two groups. Technically, you are right, it is $300$ dependent variables, but imagine that you want to predict the group membership by looking at the variables (after all, the purpose of running MANOVA is to test if the groups are different or not). Now group identity suddenly becomes a dependent variable, and you have $300$ independent variables to make the prediction.
So I think the distinction between dependent and independent variables is not very important here, and your situation can be safely described as "large $p$, small $n$".
In practice, people definitely do refer to classification problems, e.g. linear discriminant analysis, with number of features $p\gg n$ as "large $p$, small $n$" (see e.g. The Elements of Statistical Learning 18.2). But linear discriminant analysis is almost the same thing as MANOVA, see here: How is MANOVA related to LDA? So I would advocate to go ahead and to call it "large $p$, small $n$" in MANOVA context as well.
