Thanks Taco, Hans!
I did not really get what you both meant in your replies but having
added \testpage[5] before the table title seems to be doing the trick.
Is this something that can be used here?
>> One small issue appeared, though.
>> Sometimes I get title on one page and the table on the following.
>
>
> I'd opt for some judicious use of the \page command where this
> happens.
or (in the case of natural tables) try splitmethod=b or set
\extratblsplitheight to some value