You can always negotiate around whatever down - payment the advertised lease deal specifies. As for the "never put money down" line, as far as I know, there are two reasons for this, both of which have some merit but neither of which is particularly compelling.
1. Any money you pay up front is giving the financing company the benefit of the time value of that money. If your lease has an effective interest rate of 1-2%, as is often the case, you're probably better off investing that money yourself rather than effectively pre-paying a very low interest loan. Of course, this is somewhat theoretical and probably has very little real world value, given the small amounts at stake.
2. Up-front money is technically at risk in a lease if the car is totaled or stolen in the early months of the lease. The way leases are written up, you are responsible for the full payments even if the car is lost or stolen. For the most part, you're covered by insurance. The problem lies in the fact that the value of the car has significantly depreciated the day you drive it off the lot, but that significant depreciation is amortized in equal monthly amounts in a lease, so at the beginning of the lease the replacement (insurance) value of the car is less than what is reflected in the remaining payments you owe. I know I'm not stating this very clearly, but suffice it to say this risk can be covered by "gap insurance" (which they will try to sell you). Alternatively, you can do what most people do and just accept the very small risk that you total your car or it gets stolen during the first 12 months or so of the lease term.
Also, I think when you are comparing lease deals, its helps to find a baseline such as no money down, just for purposes of comparing apples to apples.
When I did my last lease - for my wife - I had an old car I wanted to trade in. It had some major engine issues and I was not comfortable selling it privately, and just didn't want that hassle. I negotiated a decent value and then had the amount applied as an upfront payment on the lease. It was only a few thousand, so was fine as a down payment. For me (and my wife), its nice from a psychological perspective to have that much lower payment each month going forward. They would have paid me in cash, but it wasn't enough money to make a difference so I just took it in the form of lower payments.