I am using Texmaker 4.5 on Windows 10 and the built-in spell-checker stops working if I copy and paste a section into the document. It seems to work/not-work on a paragraph by paragraph basis. If I try to spell-check a non-working section I am told that no misspelled words were detected, even if I have complete gibberish in the paragraph.
The figure below shows the issue I am having, see how the three gibberish words above the citation are not marked as misspelled, while the words below are correctly tagged with a red line.
Additionally, it appears that if I use \cite{} in the section that was once copied into the document the spell checker works until I close the brackets. For example, here is a working spell-checker
and here is a unworking spell-checker. 
All I did was close the bracket and the spell checker stopped working.
I have checked this the same document on my laptop running texmaker 4.1 on a Linux platform and it works perfectly.
Best Answer
I am a beginner user of LaTex, so apologies if the answer is not professional, but it solved my problem.
I had the same issue and tried to update to the latest version (5.0.2) of Texmaker but no luck. I was using {natbib} package and the command \cite{..}.
However, when I changed the citation package from \usepackage{natbib} to \usepackage{apacite} and only use \citeA{...} or citeNP{...} commands the spelling check starts to work as normal. Even with the {apacite} package the issue still appears if I use the command \cite{..}.
I hope this might encourage expert users to give a detailed answer!