We have conducted a series of laboratory simulations of earthquakes using granite cylinders containing precut bare fault surfaces at 50 MPa confining pressure. Axial shortening rates between 10-4 and 10-6 mm/s were imposed to simulate tectonic loading. Average loading rate was then modulated by the addition of a small-amplitude sine wave to simulate periodic loading due to Earth tides or other sources. The period of the modulating signal ranged from 10 to 10,000 s. For each combination of amplitude and period of the modulating signal, multiple stick-slip events were recorded to determine the degree of correlation between the timing of simulated earthquakes and the imposed periodic loading function. Over the range of parameters studied, the degree of correlation of earthquakes was most sensitive to the amplitude of the periodic loading, with weaker dependence on the period of oscillations and the average loading rate. Accelerating premonitory slip was observed in these experiments and is a controlling factor in determining the conditions under which correlated events occur. In fact, some form of delayed failure is necessary to produce the observed correlations between simulated earthquake timing and characteristics of the periodic loading function. The transition from strongly correlated to weakly correlated model earthquake populations occurred when the amplitude of the periodic loading was approximately 0.05 to 0.1 MPa shear stress (0.03 to 0.06 MPa Coulomb failure function). Lower-amplitude oscillations produced progressively lower correlation levels. Correlations between static stress increases and earthquake aftershocks are found to degrade at similar stress levels. Typical stress variations due to Earth tides are only 0.001 to 0.004 MPa, so that the lack of correlation between Earth tides and earthquakes is also consistent with our findings. A simple extrapolation of our results suggests that approximately 1% of midcrustal earthquakes should be correlated with Earth tides. Triggered seismicity has been reported resulting from the passage of surface waves excited by the Landers earthquake. These transient waves had measured amplitudes in excess of 0.1 MPa at frequencies of 0.05 to 0.2 Hz in regions of notable seismicity increase. Similar stress oscillations in our laboratory experiments produced strongly correlated stick-slip events. We suggest that seemingly inconsistent natural observations of triggered seismicity and absence of tidal triggering indicate that failure is amplitude and frequency dependent. This is the expected result if, as in our laboratory experiments, the rheology of the Earth's crust permits delayed failure.