The title is my question. I think the answer is yes, but I am unsure about it.
No, generally this will not be the case. The yields of the on-the-run treasuries for specific tenors are collected and a cubic spline is fit to these points. Thus, the yields will only match for the on-the-run 1m, 3m, 6m, 12m, 2y, 3y, 5y, 7y, 10y, and 30y treasuries.
For details please see the page on Treasury Yield Curve Methodology.
The Treasury's yield curve is derived using a quasi-cubic hermite spline function. Our inputs are the Close of Business (COB) bid yields for the on-the-run securities. Because the on-the-run securities typically trade close to par, those securities are designated as the knot points in the quasi-cubic hermite spline algorithm and the resulting yield curve is considered a par curve.
More specifically, the current inputs are the most recently auctioned 4-, 13-, 26-, and 52-week bills, plus the most recently auctioned 2-, 3-, 5-, 7-, and 10-year notes and the most recently auctioned 30-year bond, plus the composite rate in the 20-year maturity range. The quotes for these securities are obtained at or near the 3:30 PM close each trading day. The inputs for the four bills are their bond equivalent yields.