  We investigate the effects of a massive ($\gtrsim4\times10^{10}M_\odot$)
Sagittarius dwarf spheroidal galaxy (Sgr) on stellar streams using test
particle simulations in a realistic Milky Way potential. We find that Sgr can
easily disrupt streams formed more than $\sim3$ Gyr ago, while stars stripped
more recently are generally unaffected. In certain realizations, Sgr is able to
produce asymmetry between the leading and trailing tails of Pal 5,
qualitatively similar to observations. Using data from the Gaia space telescope
and elsewhere, we fit models to the GD-1 stream in the presence of a Sgr with
various initial masses. While the best-fitting models do show perturbations
resulting from interactions with Sgr, we find that the level of disruption is
not significantly greater than in the observed stream. To investigate the
general effects of Sgr on a population of streams, we generate 1000 mock
streams on GD-1-like orbits with randomized orientations. Some streams show
clear evidence of disruption, becoming folded on the sky or developing
asymmetry betweeen their two tails. However, many survive unaffected and the
peak surface brightness of stars is decreased by no more than $\sim0.3$
mag/arcsec$^2$ on average. We conclude that Sgr having an initial mass of
$\gtrsim4\times10^{10}M_\odot$ is compatible with the survival and detection of
streams formed more than 3 Gyr ago.
