The Magellanic Stream (MS) might have grown out of tidal interactions at high redshift, when the young galaxies were close together, rather than from later interactions among the Magellanic Clouds and Milky Way (MW). This is illustrated in solutions for the orbits of Local Group (LG) galaxies under the cosmological condition of growing peculiar velocities at high redshift. Massless test particles initially near and moving with the Large Magellanic Cloud in these solutions end up with distributions in angular position and redshift similar to the MS, though with the usual overly prominent leading component that the MW corona might have suppressed. Another possible example of the effect of conditions at high redshift is a model primeval stream around the LG galaxy NGC 6822. Depending on the solution for LG dynamics, this primeval stream can end up with a position angle similar to the H I around this galaxy and a redshift gradient in the observed direction. The gradient is much smaller than observed but might have been increased by dissipative contraction. Presented also is an even more speculative illustration of the possible effect of initial conditions, primeval stellar streams around M31.