Different Forms Of "Crowding Out"
1) The trade deficit can increase (and it has in fact done so in America since the "stimulus bill" was passed in February 2009), meaning that some of the increase in domestic demand will benefit foreign producers instead of domestic producers. Because the flip side of an increase in the trade deficit (or more strictly the current account deficit) is an increase in net capital inflows, this will on the other hand limit the increase in interest rates. But the point is that the "crowding out" effect will appear as an increase in the trade deficit, not an increase in interest rates.
2) To the extent that consumers acts according to the theory of Ricardian equivalence, an increase in the budget deficit will cause consumers to restrict their consumption. On the other hand, a higher savings rate will limit the increase in interest rates. But the point is that the "crowding out" effect will appear as an increase in the savings rate, not as an increase in interest rates.
3) To the extent the Fed "monetizes" the deficit by printing money, this will raise prices. The indirect effect of the deficit in this case in the form of higher prices will reduce people's purchasing power, "crowding out" the direct increase in purchasing power created by the increase in the deficit.
It is thus very misleading (in addition, there is another problematic factor in the form of other factors which would have caused interest rates to go up or down regardless of fiscal policy) to look at interest rates alone as a guide to the extent of crowding out.