Incompatible month formats between biblatex and Mendeley

The standard BibTeX abbreviations should not be given in braces, but bare as in

month = mar,

only then does BibTeX understand their special meaning.

You can read a bit about this in Tame the Beast where we find (on p. 13)

It's better having a numerical value, or an abbreviation, instead of the complete name of the month.

There it is not that clear that the abbreviation needs to be give without braces or quotation marks.

on p. 44 it says

Namely, for instance, months should be entered numerically.

So month = {jul}, is not the format recommended by Tame the Beast.

Have a look at the following BibTeX example

\documentclass{article}
\begin{filecontents}{\jobname.bib}
@article{Blavatskyy2011,
  author = {Blavatskyy, Pavlo R.},
  doi = {10.1287/mnsc.1100.1285},
  month = mar,
  number = {3},
  pages = {542--548},
  title = {{A Model of Probabilistic Choice Satisfying First-Order Stochastic Dominance}},
  volume = {57},
  year = {2011},
}

@article{bar,
  author = {{The Pierre Auger Collaboration}},
  eprint = {1107.4806},
  journal = {ArXiv e-prints},
  month = {jul},
  title = {{The Pierre Auger Observatory IV: Operation and Monitoring}},
  year = {2011}
}
\end{filecontents}

\begin{document}
These are my works, \cite{Blavatskyy2011,bar}

\bibliographystyle{plain}
\bibliography{\jobname}
\end{document}

The month wrapped in braces does not give the expected result.

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The same holds if you use biblatex with the BibTeX back-end. The new back-end Biber is so clever that it can even deal with braced month fields

So the following MWE will give the expected output (with backend=bibtex though, it won't work as expected).

\documentclass{article}
\usepackage[backend=biber]{biblatex}
\begin{filecontents}{\jobname.bib}
@article{Blavatskyy2011,
  author = {Blavatskyy, Pavlo R.},
  doi = {10.1287/mnsc.1100.1285},
  month = mar,
  number = {3},
  pages = {542--548},
  title = {{A Model of Probabilistic Choice Satisfying First-Order Stochastic Dominance}},
  volume = {57},
  year = {2011},
}

@article{bar,
  author = {{The Pierre Auger Collaboration}},
  eprint = {1107.4806},
  journal = {ArXiv e-prints},
  month = {jul},
  title = {{The Pierre Auger Observatory IV: Operation and Monitoring}},
  year = {2011}
}
\end{filecontents}
\addbibresource{\jobname.bib}

\begin{document}
These are my works, \cite{Blavatskyy2011,bar}

\printbibliography
\end{document}

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Even though Biber has no problems with the file exported from Mendeley I still consider the export faulty, since BibTeX cannot properly deal with this.

So if you cannot change the .bib file and still want to get rid of the warnings you should try and use Biber.