Yup. The incoming revenue at that point from taxes and tariffs is 'bursty', tied to quarterly periods, and won't quite cover minimal expenses for any length of time. Treasury can play some bookkeeping games, just like the last time Congress pulled this stunt (Yah. It's happened before. They even managed to balance the budget, sort of, for a couple years).
Eventually, around 6-10 weeks in, depending on exactly when in the quarterly and annual cycle we hit the limit, Treasury runs out of cups to hide the ball under. Some soldier will use ammunition, or be hurt and require medical care, or an outbreak of angry old people on the National Mall demanding their checks will require National Parks Service folks to come into work, and the jig will be up.
I actually think there would be some good entertainment value in not raising the debt ceiling, in terms of the 'instant outrage' and forced enlightenment of the populace as to what's actually covered by the federal budget. We currently run a deficit so large that cutting all 'discretionary' spending to zero doesn't quite balance the budget. If we go back to the last full year where we had a budget in place, FY 2009, we see that discretionary spending was 1.21 trillion dollars, and the deficit was 1.4 trillion dollars. 'Discretionary' spending included 515.4 billion for DoD, 145.2 billion for the Global War on Terror, 70.4 billion for the Department of Health and Human Services, and so on.
The technical default in Treasuries will cause problems in the bond markets, and should nicely raise interest rates for folks looking to buy Treasuries in the secondary market on the dip. Folks currently holding Treasuries should be prepared for a drop in value as the US notes become viewed as being fairly risky (Greece/Ireland/Portugal risky). This will barely be a speed bump for the stock market, though. Last time the market just charged on ahead through the federal government shutdown.