A scheme for generating tunable chaotic signals with controllable bandwidth was proposed and experimentally demonstrated based on two weak-resonant-cavity Fabry-Perot laser diodes, which is realized by that the chaotic signal from a weak-resonant-cavity Fabry-Perot laser diode (named as master weak-resonant-cavity Fabry-Perot laser diode) with tunable fiber Bragg grating optical feedback is unidirectionally injected into another weak-resonant-cavity Fabry-Perot laser diode (named as slave weak-resonant-cavity Fabry-Perot laser diode). The experimental results show that, through adjusting the central wavelength of tunable fiber Bragg grating and feedback strength, master weak-resonant-cavity Fabry-Perot laser diode can output chaotic signal whose central wavelength can be tuned within the tunable range of tunable fiber Bragg grating. Further injecting the chaotic signal from master weak-resonant-cavity Fabry-Perot laser diode into slave weak-resonant-cavity Fabry-Perot laser diode, the bandwidth of the tunable chaotic signal can be adjusted within a large range by varying the injection power and detuning frequency.