time for a signal after a sum
I have a signal s(t) of duration T with limited bandwidth.


No, you don't. 

A signal cannot be finite in both time and frequency at the same time. So the whole question about length in both domains comes down to how exactly you define it. Typically you need a criteria like "less than -100dB", "99.99% of the entire energy" or something like this. 

These criteria are specific to your application. Maybe -40dB is plenty, maybe you need more than -200dB. Once you have a criteria defined you can apply it to your signals and determine the "lengths", but the result will always be tied to an application specific definition.