Is this better?

just add \tabulinesep=_3pt^3pt:
\documentclass{article}
\usepackage{geometry} % <-- added
\usepackage{tabu}
\tabulinesep=_3pt^3pt% <-- added
\begin{document}
\begin{tabu} to \linewidth {*6{X[c]}}
\tabucline{-}
sensor molecule & position & chemical shift reference & some & other &info \\
\tabucline{-}
xyz & C$_1$ & aaa & 1.11 & 3.54 & 11.00 \\
xyz & C$_5$ & bbb & 2.22 & 44.54 & 12.0 \\
\tabucline{-}
\end{tabu}
\end{document}
Off topic: in your table is not need to manually break cell contents, this will done by column type X, if necessary.
Edit:
Source of not centering third column header is to wide content. If you make \textwidth longer, for example with adding of package geometry, the content will become correctly centered.
Addendum:
Less problems is with tabularx and use booktabs for table's rules. With siunitx you can align numbers at decimal points. For equal width of all columns is used trick with multicolumn environment for S column headers:

\documentclass{article}
\usepackage{geometry}
\usepackage{booktabs,tabularx}
\newcolumntype{C}{>{\centering\arraybackslash}X}
\newcommand\mcx[1]{\multicolumn{1}{C}{#1}}
\usepackage{siunitx}
\begin{document}
\begin{tabularx}{\linewidth}{*{3}{C}
*{3}{S[table-format=2.2]}}
\toprule
sensor molecule & position & chemical~shift reference
& \mcx{some} & \mcx{other} & \mcx{info} \\
\midrule
xyz & C$_1$ & aaa & 1.11 & 3.54 & 11.00 \\
xyz & C$_5$ & bbb & 2.22 & 44.54 & 12.0 \\
\bottomrule
\end{tabularx}
\end{document}
\makecellcommand available? – Andrew Swann Mar 10 '17 at 12:11