We present a general analysis of the asymptotic behaviour of the soft-constrained time-optimal (SCTO) control of linear systems. We are essentially seeking to show that (1) this control is bounded, like its hard-constrained counterpart; and (2) there is a basic tradeoff between the control bounds and the speed of the response, and we seek to characterize this tradeoff. We show that if the 'No Infinite Manoeuvre in Finite Optimal Time' (NIMIFOT) condition holds, then the boundedness of the SCTO control can be easily checked, and that the above-mentioned tradeoff is clearly characterized. For instance, we show that the SCTO control is bounded for all stable systems and for many marginally stable systems provided that the NIMIFOT condition holds. We also show that a slight modification of the formulation of the SCTO control problem can ensure that the NIMIFOT condition holds. With this new formulation, the SCTO control can be shown to be bounded for most linear plants satisfying mild and easily checkable conditions.