Janice -

That comment sounds like it might be a software limitation. If you saw it in a general reference to stratified random sampling, not a specific software package, then it sounds like someone's idea of how likely it would be that you found more than six good categories, but I'd think you certainly could.

Something else comes to mind about your application, however. Do you want to find results overall only, or by each university? The idea behind a stratified random sample is to efficiently find overall results. You will obtain results by stratum also, but some may have high standard errors. If you want good results for each stratum, you will not be doing optimum allocation for a stratified random sample. If you want good results for each university, your sample size will likely be larger. You would actually want to do a separate simple random sample for each of the eight subpopulations.

Actually you might want to look at the university structure and either do a simple random sample in each, or in each, do a stratified random sample, or a cluster sample, multistage sample, whatever is appropriate. You might want to look at a survey sampling book, such as one or more of the following:

Thompson, S.K.(2012), Sampling, 3rd ed, John Wiley & Sons.

Cochran, W.G.(1977), Sampling Techniques, 3rd ed, John Wiley & Sons. 

Blair, E. and Blair, J(2015), Applied Survey Sampling, Sage Publications.

Lohr, S.L(2010), Sampling: Design and Analysis, 2nd ed., Brooks/Cole.

If you had auxiliary or regressor data I would add to this list of suggested textbooks, but I don't suppose that is the case. The four examples above have a variety of topics at more than one level, so you could explore that. Some styles are easier for some people to understand than others.

Best wishes - Jim

PS - Silly me. You likely need an online reference considering there's a pandemic! Penn State has a lot of good information out there. I just searched and found two possibly helpful URLs:

https://online.stat.psu.edu/stat100/lesson/2/2.4

and

https://online.stat.psu.edu/stat506/node/27/

If you search with "Penn State" and a topic, you can find a great deal of good information.

I don't think so; but I know that increasing the number of strata in a survey has something to do with cost implication. In addition, using faculties as strata depends on your variable of interest and the factor used to stratify. They can also be used as clusters. I hope you know the differences between strata and clusters. It is in cluster sampling that I know that the size of clusters has effect on the precision of estimates.

Thank you James R Knaub and Emmanuel Ekpenyong for the information. It help me a lot.