Confusion about the meaning of Bandwidth-limited pulse
The more correct term is transform limited. That refers to a pulse that has the shortest possible duration for it's bandwidth.
Not all pulses are transform limited. Think of a burst of static for example. It has many random fluctuations up and down. Compare to a short gaussian pulse that rises up smoothly and then back down. If both have the same bandwidth, the latter pulse will be much, much shorter. This process can be understood formally using Fourier analysis, but that is the basic idea.