Breakup mechanisms for chirped picosecond pulse in the single-mode fibers are numerically analyzed through solving nonlinear Schrdinger equation by the standard split-step Fourier method. The results show that, the breakup of shorter chirped pulse (10 ps) is caused by pulse collapse due to the high-order soliton compression, while the breakup of longer chirped pulse (200 ps) is caused by the nonlinear amplification of noise due to the modulation instability. For chirped pulse of intermediate durations (50 ps), its breakup is caused by modulation instability instead of high-order soliton compression due to the role of noise. Initial positive and negative chirps can speed and slow down the process of shorter pulse breakup, respectively. However, the initial chirp has little influence on the breakup of longer pulses. The effect of initial chirp on pulse breakup is closely related to the breakup mechanisms for pulse.