Let's say NX = 0.
Y=C+I+G
Let's also assume the government doesn't borrow: G = T
Obviously I+C=Y-T => I+C=Y-G
In this case, every increase in G does crowd out C+I, because if the government's only source of funding is taxation, every increase in G has to come from an increase in T, which lower C+I.
If the government can borrow money, however, it depends from whom it borrows. If it borrows only from people within the country, the demand for money goes up (a shift to the right) interest rates go up and overall funds loaned out increase, while private investment decreases (the private demand for money didn't change but the additional governmental demand created higher equilibrium interest rate, at which private businesses want to invest less). Also, since it's only borrowing from within the country, every Dollar loaned to the government has to be forgone consumption today (or a loan a private business didn't get, as explained earlier).
So as long as the Y=C+I+G identiy exists, every Dollar the government spends, is indeed a Dollar not spent by private individuals.
Only if we look at foreign debt, the picture changes a little bit. Norway for example exploited huge oil reserves and thus was attractive for investors all over the world. However, once the loans are repaid, the investors want to consume eventually. If there are many currency holders from outside the country wanting to consume goods, this obviously decreases domestic consumption (higher prices, etc.). That's not problematic in Norway's case, since they actually invested in something of real value to consumers, so they are still better off than without the loans.
The US is a little different because the Dollar is a reserve currency. Many foreign lenders hold Dollar-denominated debt just for the sake of holding it and keep renewing it, without any imidiate interest in consuming products made in the US. This is btw one the major reasons consumer prices in the US rise slower than asset prices. The US was not an attractive investment destination because of it's promising capital accumulating projects. In fact, the US blew all that additional money on consumption, either by the government or private consumption (just look at the enormous trade defict). And the investment projects they did engage in are not needed and have never been sustainable, for reasons already explained. At some point foreign investors are not going to invest into US bonds any longer. Not only is this going to be abysmal for the US treasury, causing the Fed most likely to expand the money supply even further, but that's going to mean a forced shift in the trade balance.