This forum won't let me upload a .lyx file!?!? Anyway, here is the source.
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% Preview source code
%% LyX 2.0.0 created this file. For more info, see http://www.lyx.org/.
%% Do not edit unless you really know what you are doing.
\documentclass[english]{article}
\usepackage[T1]{fontenc}
\usepackage[latin9]{inputenc}
\usepackage[letterpaper]{geometry}
\geometry{verbose,tmargin=1in,bmargin=1in,lmargin=1in,rmargin=1in}
\setlength{\parskip}{\bigskipamount}
\setlength{\parindent}{0pt}
\usepackage{array}
\makeatletter
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% LyX specific LaTeX commands.
%% Because html converters don't know tabularnewline
\providecommand{\tabularnewline}{\\}
\makeatother
\usepackage{babel}
\begin{document}
This document is setup as a US letter size, with 1'' margins all around.
The table below is set to have a table-wide setting table width of
100\% of the text width. The far ends of the table do line up to 100\%
of the text width, but some other very strange things happen, such
as part of the top border cut off and the text on the first row always
aligned to the right.
\begin{tabular*}{1\textwidth}{@{\extracolsep{\fill}}|c|c|c|}
\cline{2-3}
\multicolumn{1}{c|}{} & This is a test & test column\tabularnewline
\hline
123 mph & & \tabularnewline
\hline
345 mph & & \tabularnewline
\hline
\end{tabular*}
The table below uses no table-wide settings, and columns that are
setup as 33\% of the text width, but for some reason the columns do
not render to 33\% of the text width. You will see when this file
is rendered to PDF, PS, etc...
\begin{tabular}{|>{\centering}p{0.33\textwidth}|>{\centering}p{0.33\textwidth}|>{\centering}p{0.33\textwidth}|}
\cline{2-3}
\multicolumn{1}{>{\centering}p{0.33\textwidth}|}{} & This is a test & test column\tabularnewline
\hline
123 mph & & \tabularnewline
\hline
345 mph & & \tabularnewline
\hline
\end{tabular}
How do I fix this problem?
\end{document}