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Condition 1 implies that both $\|P_{[q_2]}y\|$ and $\|P_{[\tilde{q}_2]}y\|$ are big (whence $\|P_{[q_2, q_3]}y\| \geq \|P_{[q_2]}y\|$ is big), while condition 2 implies both $\|P_{[q_3]}y\|$ and $\|P_{[\tilde{q}_3]}y\|$ are small. While the first implication is straightforward, the second implication needs elaboration: without loss of generality, let's show that $\|P_{[q_3]}y\|$ is small. To this end, note that $[e, x_2] = [q_1, q_2]$ and $[e, x_2, x_1] = [q_1, q_2, q_3]$, it follows that \begin{align} \|P_{[q_3]}y\|^2 = \|P_{[e, x_2, x_1]}y\|^2 - \|P_{[e, x_2]}y\|^2 = \|P_{[e, x_2]^\perp}\left(P_{[e, x_2, x_1]}y\right)\|^2, \end{align}
where $[e, x_2]^\perp$ is the orthogonal complement of $[e, x_2]$. Under condition 2, it is easy to see that $[e, x_2, x_1] \approx [e, x_2]$, whence the projection of $P_{[e, x_2, x_1]}y$ onto $[e, x_2]^\perp$ is very close to zero, hence $\|P_{[q_3]}y\|^2$ must be small. Similarly, $\|P_{[\tilde{q}_3]}y\|^2$ is small. In summary, we showed that under condition 1 and condition 2, $\eqref{5}$ is big (hence $F_0$ in $\eqref{4}$ is big, which results in significant $F$-test outcome) and $\eqref{6}$ and $\eqref{7}$ are small (hence $F_1$ in $\eqref{3.1}$ and $F_2$ in $\eqref{3.2}$ are small, which results in insignificant $t$-test outcome).

Condition 1 implies that both $\|P_{[q_2]}y\|$ and $\|P_{[\tilde{q}_2]}y\|$ are big (whence $\|P_{[q_2, q_3]}y\| \geq \|P_{[q_2]}y\|$ is big), while condition 2 implies both $\|P_{[q_3]}y\|$ and $\|P_{[\tilde{q}_3]}y\|$ are small. While the first implication is straightforward, the second implication needs elaboration: without loss of generality, let's show that $\|P_{[q_3]}y\|$ is small. To this end, note that $[e, x_2] = [q_1, q_2]$ and $[e, x_2, x_1] = [q_1, q_2, q_3]$, it follows that \begin{align} \|P_{[q_3]}y\|^2 = \|P_{[e, x_2, x_1]}y\|^2 - \|P_{[e, x_2]}y\|^2 = \|P_{[e, x_2]^\perp}\left(P_{[e, x_2, x_1]}y\right)\|^2, \end{align}
where $[e, x_2]^\perp$ is the orthogonal complement of $[e, x_2]$. Under condition 2, it is easy to see that $[e, x_2, x_1] \approx [e, x_2]$, whence the projection of $P_{[e, x_2, x_1]}y$ onto $[e, x_2]^\perp$ is very close to zero, hence $\|P_{[q_3]}y\|^2$ must be small. Similarly, $\|P_{[\tilde{q}_3]}y\|^2$ is small. In summary, we showed that under condition 1 and condition 2, $\eqref{5}$ is big (hence $F_0$ in $\eqref{4}$ is big, which results in significant $F$-test outcome) and $\eqref{6}$ and $\eqref{7}$ are small (hence $F_1$ in $\eqref{3.1}$ and $F_2$ in $\eqref{3.2}$ are small, which results in insignificant $t$-test outcomes).

The key concept in the test statistic $\eqref{3.1}$ is the term $SSR(X_1|X_2)$, known as sequential/extra sums of squares (see Chapter 7 of the same reference above and this link for more details), which accounts for the reduction in the error of sum caused by adding $X_1$ to the regression model when $X_1$ is already in the model. Thus $SSR(X_1|X_2)$ measures the marginal effect of adding $X_1$ to the regression model when $X_1$ is already in the model. Clearly, $SSR(X_2|X_1)$ bears the same interpretation. Having understood this concept, comparing $\eqref{3.1}$-$\eqref{3.2}$ against $\eqref{4}$ immediately indicates a scenario such that the overall $F$-test is significant while partial $F$-tests are insignificant, namely when $SSR(X_1, X_2)$ is big but both $SSR(X_1|X_2)$ and $SSR(X_2|X_1)$ are small (relative to $MSE$). In other words, the variation of $y$ can be adequately explained by any single predictor $X_1$ or $X_2$ (therefore $SSR(X_1, X_2)$ is big), but when we tried to explain the variation of $y$ by using both $X_1$ and $X_2$, the further improvement is not significant (therefore both $SSR(X_1|X_2)$ and $SSR(X_2|X_1)$ are small). One concrete example of this scenario is multicollinearity, as described in @Rob Hyndman's answer. However, this is not the only cause of the posed paradox, and we will come back to another scenario later.

The key concept in the test statistic $\eqref{3.1}$ is the term $SSR(X_1|X_2)$, known as sequential/extra sums of squares (see Chapter 7 of the same reference above and this link for more details), which accounts for the reduction in the error sum of squares caused by adding $X_1$ to the regression model when $X_2$ is already in the model. Thus $SSR(X_1|X_2)$ measures the marginal effect of adding $X_1$ to the regression model when $X_2$ is already in the model. Clearly, $SSR(X_2|X_1)$ bears the same interpretation. Having understood this concept, comparing $\eqref{3.1}$-$\eqref{3.2}$ against $\eqref{4}$ immediately indicates a scenario such that the overall $F$-test is significant while partial $F$-tests are insignificant, namely when $SSR(X_1, X_2)$ is big but both $SSR(X_1|X_2)$ and $SSR(X_2|X_1)$ are small (relative to $MSE$). In other words, the variation of $y$ can be adequately explained by any simple regression model where the single predictor is $X_1$ or $X_2$ (therefore $SSR(X_i)$ is big, and as a result $SSR(X_1, X_2)$ is big), but when we tried to further increase the $y$-variation explain ratio by adding the remaining variable to the existing simple regression model, the improvement is very limited (therefore both $SSR(X_1|X_2)$ and $SSR(X_2|X_1)$ are small). One example that fits this setting is multicollinearity, as described in @Rob Hyndman's answer. However, multicollinearity is not the only example that fits the general setting, as I will briefly discuss at the end of this answer.

Another cause of this paradox is as described by the second reason of Jeromy Anglim's answer, and can also be explained using the framework built earlier. This actually corresponds to the scenario \begin{align} & \max(\|P_{[q_3]}y\|^2, \|P_{[\tilde{q}_3]}y\|^2) \leq MSE \cdot q_1^*, \\ & \|P_{[q_2]}y\|^2 + \|P_{[q_3]}y\|^2 > 2MSE \cdot q_2^*. \tag{8}\label{8} \end{align} Since we have the freedom of controlling the magnitudes of $P_{[q_2]}y$ and $P_{[q_3]}y$ (from the counterexample construction perspective), there are countless concrete examples such that $\eqref{8}$ holds.

Another cause of this paradox is as described by the second reason of Jeromy Anglim's answer, and can also be explained using the framework built earlier. This actually corresponds to the scenario \begin{align} & \max(\|P_{[q_3]}y\|^2, \|P_{[\tilde{q}_3]}y\|^2) \leq MSE \cdot q_1^*, \\ & \|P_{[q_2]}y\|^2 + \|P_{[q_3]}y\|^2 > 2MSE \cdot q_2^*. \tag{8}\label{8} \end{align} Since we have the freedom of controlling the magnitudes of $P_{[q_2]}y$ and $P_{[q_3]}y$ (from the counterexample construction perspective), for $q_1^* < 2q_2^*$, there are countless concrete examples such that $\eqref{8}$ holds.