The Late Heavy Bombardment happened about 4 billion years ago, and lasted for a couple hundred million years. The prevailing theory is that it occurred when the outer giant planets changed their orbits, scattering many small bodies from the asteroid and Kuiper belts. This dramatically changed the structure of the outer Solar System. Many of these small bodies - asteroids, for example - hit the terrestrial planets, as well as the Moon.
We do appear to have a second case of this happening, in the planetary system of Eta Corvi. Some of the dust in the system is theorized to have come from collisions between asteroids and similarly-sized bodies, as well as from a planetary-mass object. The trigger for the bombardment is unknown, although it could be due to planetary migration, as in our own Solar System.
That said, the bombardment rates during the Late Heavy Bombardment are much lower than what you're looking for - likely because there just isn't enough matter in the Solar System to keep up impacts at the rate you want for long geological timescales. On shorter timescales, though, a drastically sped-up bombardment could work.
For a nice visualization, check out this simulation on YouTube.
At $t=505$ million years, the system is pretty stable. The gas giants are, in order, Jupiter, Saturn, Neptune, and Uranus, and the green bodies represent objects in the Kuiper Belt. The inner terrestrial planets are not shown:
At $t=882$ million years, the system has recently experienced an instability. The orbits of Neptune and Uranus are changing, and the smaller objects are being scattered across the Solar System:
At $t=983$ million years, the giant planets have started to settle down a bit, and better resemble the outer Solar System as seen today. The Kuiper Belt is now substantially less dense than it originally was - more like its current structure:
For a comparison, here's Figure 8 of Lisse et al. (2012), showing the modeled structure of the Eta Corvi system. The red bodies are the analog of the Kuiper Belt; the inner blue bodies constitute a smaller disk that may be made of debris from a planet that sustained extreme collisions with asteroids: