Summary: We investigate through analysis and computational experiment explicit second- and third-order strong-stability preserving (SSP) Runge-Kutta time discretization methods in order to gain perspective on the practical necessity of the SSP property. We consider general theoretical SSP limits for these schemes and present a new optimal third-order low-storage SSP method that is SSP at a Courand-Friedrichs-Levy (CFL) number of 0.838. We compare results of practical preservation of the total variation diminishing (TVD) property using SSP and non-SSP time integrators to integrate a class of semi-discrete Godunov-type spatial discretizations. Our examples involve numerical solutions to Burgers' equation and the Euler equations. We observe that well-designed non-SSP and non-optimal SSP schemes with SSP coefficients less than one provide comparable stability when used with time steps below the standard CFL limit. Results using a third-order non-TVD central weighted essentially-nonoscillatory (CWENO) scheme are also presented. We verify that the documented SSP methods with the number of stages greater than the order provide a useful enhanced stability region. We show by analysis and by numerical experiment that the non-oscillatory third-order reconstructions used by \textit{X.-D. Liu} and \textit{E. Tadmor} [Numer. Math. 79, No. 3, 397-425 (1998; Zbl 0906.65093)] and by \textit{A. Kurganov} and \textit{G. Petrova}, ibid. 88, No. 4, 683--729 (2001; Zbl 0987.65090)] are in general only second- and first-order accurate, respectively.