In this paper, we consider the estimation of the unknown parameters of the multiple chirp signal model in presence of additive error. The chirp signals are quite common in many areas of science and engineering, specially sonar, radar, audio signals etc. The observed signals are usually corrupted by noise. In different signal processing applications it is observed that the errors may be heavy tailed. In this paper it is assumed that the additive errors have mean zero but may not have finite variance and are independent and identically distributed. We consider the least squares estimators and the approximate least squares estimators which maximize a periodogram like function. It has been observed that both the estimators are strongly consistent. The asymptotic distribution of the least squares estimators is obtained under the assumption that the additive errors are from a symmetric stable distribution. The approximate least squares estimators have the same asymptotic distribution as the least squares estimators. We perform some numerical simulations to see how the proposed estimators work. It is observed that the least squares estimators perform slightly better than the approximately least squares estimators in terms of the biases and mean absolute deviation.