With liberal use of the \multicolumn{.}{.}{...} command, you can get away with spreading the table across the entire \linewidth:
\documentclass{article}
\usepackage{array,booktabs}
\begin{document}
\begin{table}[ht]
\centering
\caption{An interesting table}
\label{tbl:interesting}
\begin{tabular}{*{6}{p{.16\linewidth}}}
\multicolumn{6}{c}{Panel A: Some stuff} \
\toprule
\multicolumn{2}{p{.33\linewidth}}{First name} & \multicolumn{2}{p{.33\linewidth}}{Last name} &
\multicolumn{2}{p{.33\linewidth}}{Product} \
\midrule
\multicolumn{2}{l}{Bubba} & \multicolumn{2}{l}{Gump} & \multicolumn{2}{l}{Shrimp} \
\multicolumn{2}{l}{Steve} & \multicolumn{2}{l}{Jobs} & \multicolumn{2}{l}{Happiness} \
\bottomrule
\
\multicolumn{6}{c}{Panel B: Other stuff} \
\toprule
\multicolumn{3}{p{.49\linewidth}}{School} & \multicolumn{3}{p{.49\linewidth}}{State} \
\midrule
\multicolumn{3}{l}{Harvard} & \multicolumn{3}{l}{MA} \
\multicolumn{3}{l}{Yale} & \multicolumn{3}{l}{CT} \
\multicolumn{3}{l}{Brown} & \multicolumn{3}{l}{RI} \
\bottomrule
\end{tabular}
\end{table}
\end{document}
Since the two panels are contained in the same tabular, they span the same width. The above uses the booktabs package for presentation of the tabular environments. However, it is not necessarily needed. If you want to drop it, you should also drop/replace the \toprule, \midrule and \bottomrule rules with \hline or another preference.

Alternatively, you could also use the tabularx package to spread columns across a specific width:
\documentclass{article}
\usepackage{booktabs,tabularx}
\begin{document}
\begin{table}[ht]
\centering
\caption{An interesting table}
\label{tbl:interesting}
\medskip
\begin{tabularx}{\linewidth}{ X X X }
\multicolumn{3}{c}{Panel A: Some stuff} \
\toprule
First name & Last name & Product \
\midrule
Bubba & Gump & Shrimp \
Steve & Jobs & Happiness \
\bottomrule
\end{tabularx}
\bigskip
\begin{tabularx}{\linewidth}{ X X }
\multicolumn{2}{c}{Panel B: Other stuff} \
\toprule
School & State \
\midrule
Harvard & MA \
Yale & CT \
Brown & RI \
\bottomrule
\end{tabularx}
\end{table}
\end{document}

subfigurepackage lets you stack tables vertically, for example by placing a \ or\parbetween the\subfigures. – Aug 23 '11 at 16:15\textwidth, that looks terrible. Just compare the output of Stefan's solution (not spread) with Werner's's (spread). Using thebooktabspackage is a good idea though. – doncherry Aug 23 '11 at 17:35