4x5870 would be no problem, 5x5870 will be cutting it a bit close. Don't get me wrong, it's a very good PSU (it supplies 80A on a single 12V rail, basically perfect for a mining rig), it's just that 5 5870s will push even that close to the limit. But if you optimize your power consumption (underclock your cpu, underclock the memory on all video cards, don't overvolt the GPUs), it should just about handle it. Still, for five 5870s, you're probably better off playing it safe and getting a 1.2kW unit.
As usual everyone vastly over estimates needed power. Running a 4x5870 rig and a Kill-a-watt monitor I measured my rig to pull about 900Watts from the wall on a gold standard PSU, which means about 785Watts to the system.
So any quality 800 - 850Watt PSU will do you fine, just make sure you get something with either a single rail, or rails that can handle the amperage you draw for the GPUs.
How is estimating about 200W per 5870 in a 4-way rig 'overestimating' needed power? There's more to power supplies than just the number printed on the box. What really matters is the 12V rail, which will supply close to 100% of the power the video cards will draw. A 1kW PSU with a 36A 12V rail is just plain crap and it will actually be dangerous to use it in a 3x or 4x 5870 rig. There's a realistic danger of something in the PSU blowing up, melting, etc., possibly causing damage to attached components or even starting a fire.
That said, a good 850W PSU (70A @ single 12V rail) can handle 4 5870s, if you optimize power consumption and don't overdo it with GPU overclocking. A 1kW unit is still a safer bet though.
An 85% efficient 850W would give ~722W to play with, minus 70W for system leaves ~652W for GPUs. ~652W/4 = 163W per GPU available.
That's not how PSU efficiency is measured. The rated Wattage is what the PSU can output, not what it will draw from the socket. So, an 850W PSU will actually give you ~800-850W to play with, depending on the rail configuration. Again, the important one is the 12V rail, which in a typical quality 850W unit will deliver 70A or thereabout, so ~840W.