EDIT
I was given the following code for a table template (modified in order to take into account David's remarks below) :
\documentclass[12pt, a4paper, final]{article}
\usepackage{array}
\usepackage{multirow}
\usepackage{booktabs}
% This command computes and creates a vertical space
% depending on the number of rows to compensate for.
% It makes use of the counter verticalcompensationrows
% and the length \verticalcompensationlength which equals
% \aboverulesep plus \belowrulesep
\newlength{\verticalcompensationlength}
\setlength{\verticalcompensationlength}{\aboverulesep}
\addtolength{\verticalcompensationlength}{\belowrulesep}
\newcounter{verticalcompensationrows}
\newcommand{\verticalcompensation}[1]{%
\setcounter{verticalcompensationrows}{#1}%
\addtocounter{verticalcompensationrows}{-1}%
\vspace*{-\value{verticalcompensationrows}\verticalcompensationlength}%
}
% This command reimplements \multirow to compensate
% for the vertical offset, but looses some functionality
% of the \multirow command (not discussed here).
\newcommand{\multirowbt}[3]{%
\multirow{#1}{#2}{\verticalcompensation{#1}#3}%
}
\begin{document}
\begin{tabular}{lp{13cm}}
\toprule
\multicolumn{2}{c}{Table title or something else ...} \\
\midrule
\multirowbt{4}{*}{My name} & A bit of text \\
\cmidrule{2-2}
& A bit more text A bit more text A bit more text A bit more text \\
\cmidrule{2-2}
& A bit more text A bit more text A bit more text A bit more text \\
\cmidrule{2-2}
& A bit more text A bit more text A bit more text A bit more text \\
\midrule
\multirowbt{3}{*}{Your name} & Some text \\
\cmidrule{2-2}
& More text More text More text \\
\cmidrule{2-2}
& More text More text More text \\
\bottomrule
\end{tabular}
\end{document}
It outputs the table shown in Figure 1 below :

However, when I have some text that covers more than one line, the vertical aligned in column 1 is no longer OK, as demonstarated by TABLE 2.
In his answer below, David gives a solution to this problem using \raisebox. This works, but it requires me to change the amount the box has to be shifted for each column of each table.
I was wondering whether anyone knew of a way to get Latex to center this content automatically.



