! Paragraph ended before name was complete

If I remove all the (wrong) braces around family names, there is no issue.

documentclass[11pt]{article}
usepackage[utf8]{inputenc}
usepackage{filecontents}

begin{filecontents}{jobname.bib}
@ARTICLE{2013APh....50...26A,
  author = {Abeysekara, A. U. and Alfaro, R. and Alvarez, C. and 
            {'A}lvarez, J. D. and Tollefson, K. and Torres, I. and
            Ukwatta, T. N. and Villase{ n}or, L. and Weisgarber, T. and
            Westerhoff, S. and Wisher, I. G. and
            Wood, J. and Yodh, G. B. and Younk, P. W. and Zaborov, D. and
            Zepeda, A. and Zhou, H.},
  title = {Sensitivity of the high altitude water {Cherenkov} detector to 
           sources of multi-{TeV} gamma rays},
  journal = {Astroparticle Physics},
  archivePrefix = {arXiv},
  eprint = {1306.5800},
  primaryClass = {astro-ph.HE},
  keywords = {TeV gamma-ray astronomy, Water cherenkov, Cosmic ray},
    year = 2013,
  month = dec,
  volume = 50,
  pages = {26-32},
  doi = {10.1016/j.astropartphys.2013.08.002},
  adsurl = {http://adsabs.harvard.edu/abs/2013APh....50...26A},
  adsnote = {Provided by the SAO/NASA Astrophysics Data System}
}
end{filecontents}

usepackage[
  language=english,
  style=numeric-comp,
  sorting=none,
%  backend=biber,
  natbib=true,
  doi=false,
%  bibencoding=utf8,
  isbn=false
]{biblatex}
addbibresource{jobname.bib}

begin{document}
This is a simple document~cite{2013APh....50...26A}

printbibliography
end{document}

Also wrong are the ~ between initials. The argument to addbibresource should contain the .bib extension.

Egreg's recommendations are very good, but it is easy to understand your situation. I tried to narrow down what was causing the problem and, as it turns out, it is one of the names {{'A}lvarez}. Somehow, that is resulting in a .bbl file with unbalanced braces, which gives you problems in your compilation. Once you are using bibencoding=utf8 you can change it to {Álvarez} and get rid of the error. I don't have your other problematic cases at hand, but my guess is the doble braces at the start cause the problem (even though I don't understand why they should cause a problem at all). If this is the case, the number of instances for you to fix would be already much, much smaller.

documentclass[11pt]{article}
usepackage{filecontents}
begin{filecontents}{jobname.bib}
@ARTICLE{2013APh....50...26A,
  author = {{Abeysekara}, A.~U. and {Alfaro}, R. and {Alvarez}, C. and {Álvarez}, J.~D. and {Tollefson}, K. and {Torres}, I. and {Ukwatta}, T.~N. and {Villase{~n}or}, L. and {Weisgarber}, T. and {Westerhoff}, S. and {Wisher}, I.~G. and {Wood}, J. and {Yodh}, G.~B. and {Younk}, P.~W. and {Zaborov}, D. and {Zepeda}, A. and {Zhou}, H.},
  title = "{Sensitivity of the high altitude water Cherenkov detector to sources of multi-TeV gamma rays}",
  journal = {Astroparticle Physics},
  archivePrefix = "arXiv",
  eprint = {1306.5800},
  primaryClass = "astro-ph.HE",
  keywords = {TeV gamma-ray astronomy, Water cherenkov, Cosmic ray},
    year = 2013,
  month = dec,
  volume = 50,
  pages = {26-32},
  doi = {10.1016/j.astropartphys.2013.08.002},
  adsurl = {http://adsabs.harvard.edu/abs/2013APh....50...26A},
  adsnote = {Provided by the SAO/NASA Astrophysics Data System}
}
end{filecontents}
usepackage[language=english, style=numeric-comp, sorting=none, backend=biber, natbib=true, bibencoding=utf8, doi=false, isbn=false]{biblatex}
addbibresource{jobname.bib}

begin{document}
This is a simple documentcite{2013APh....50...26A}
end{document}

had the same problem, figured out now: you shall not nest braces 3+ levels deep.

as @gusbrs notices it has to do with the names--but only because noting accented letters in braces needs another level of nesting.

so, author = {{ÖKO} Friendly Company}, would work, but author = {{{"O}KO} Friendly Company}, would not anymore.

even if the braces are balanced, there's an error. so author = {{{a}}}, will work, but author = {{{a} {b}}}, will not.

The last notation will throw an ! Paragraph ended before name was complete. error.

Try to remove as much braces as possible, use utf8 encoding as already mentioned, should be enough for your entries.

N.B.: biber --tool -V *.bib will not warn you about that.

Just for people with the same problem: Download the newest biber version from sourceforge (2.11 did the trick for me) and install it.

On linux, you just have to unpack the zip-file and move the biber executable to the appropriate place:

which biber
  /usr/bin/biber

sudo mv /usr/bin/biber /usr/bin/biber.old

sudo mv ./biber /usr/bin/biber

That fixed the problem for me. If I find some problems associated to this partial update down the line, I will report them here.

Just an update for what helped me (because none of the previous answers didn't): there are required fields, when you define your bibliography.bib file (or another file), in particular I was missing volume= in one of the @article's, which caused biber to generate some inappropriate structures.

TLDR: keep an eye on defining all required fields in bibliography, even if You don't need them (You can always write volume="").