Rapid accretion of cold intergalactic gas plays a crucial role in getting gas into galaxies. It has been suggested that this gas accretion proceeds along narrow streams that might also directly drive the turbulence in galactic gas, dynamical disturbances and bulge formation. In cosmological simulations, however, it is impossible to isolate and hence disentangle the effect of cold stream accretion from internal instabilities and mergers. Moreover, in most current cosmological simulations, the phase structure and turbulence in the interstellar medium (ISM) arising from stellar feedback are treated in an approximate (subgrid) manner, so that the feedback cannot generate turbulence in the ISM. In this paper we therefore test the effects of cold streams in extremely high-resolution simulations of otherwise isolated galaxy discs using detailed models for star formation and stellar feedback; we then include or exclude mock cold flows falling on to the galaxies, with mass accretion rates, velocities and flow geometry set to maximize their effect on the gaseous disc. We find (1) turbulent velocity dispersions in gas discs are identical with or without the presence of the cold flow; the energy injected by the flow is efficiently dissipated where it meets the disc. (2) In runs without stellar feedback, the presence of a cold flow has essentially no effect on runaway fragmentation (local collapse), resulting in star formation rates (SFRs) that are an order-of-magnitude too large. (3) Model discs in runs with both explicit feedback and cold flows have higher SFRs, but only insofar as they have more gas. (4) Because the flows are extended, relative to the size of the disc, they do not trigger strong resonant responses and so induce weak gross morphological perturbation (bulge formation via instabilities/fragmentation is not accelerated). (5) However, flows can thicken the disc by direct contribution of out-of-plane or misaligned star-forming streams/filaments. We conclude that while inflows are critical over cosmological time-scales to determine the supply and angular momentum of gas discs, they have weak instantaneous dynamical effects on galaxies.