Bound R-Squared for Individual and Combined OLS Regressions

What values can R-squared take in ordinary least squares, and why? On the same response and data rows, suppose the regressions y ~ x1 and y ~ x2 have R-squared values of 0.1 and 0.2. What is the possible range for y ~ x1 + x2?

Constraints & Assumptions

For the core problem, work with in-sample OLS that includes an intercept, a non-constant response, and the usual centered definition $R^2 = 1 - \frac{SSE}{SST}$. All models are fitted on identical rows. Also discuss what changes when the intercept is omitted or when predictions are made on out-of-sample data.

Clarifying Questions

Is an intercept included? Are the data and response exactly the same across all fits? Is this the ordinary R-squared or the adjusted version? Is the evaluation in-sample?

What a Strong Answer Covers

Projection or nested-model reasoning to establish the bounds, the role that correlation among predictors plays, and why knowing the two separate scores does not uniquely determine the combined score.

Follow-up Questions

Can the joint model explain all the variation even when each predictor individually has a low R-squared? Can adding a predictor ever lower the ordinary training R-squared? Under what conditions can R-squared become negative or undefined? Overview: Derive the in-sample OLS R-squared bounds and the range of possible joint-model R-squared given the individual scores, covering the intercept assumption, predictor correlation, and out-of-sample exceptions.