With $\textbf{F}(\theta)$ denoting the Fisher matrix, $\textbf{V}(\theta)$ the variance matrix, $\textbf{C}(\theta)$ the covariance matrix, and parameter vector $\theta = [\theta_1, \theta_2, \dots, \theta_N]$. It is well known that the covariance matrix
$$\textbf{C}(\theta) \ge \textbf{F}^{-1}(\theta) ~~~~~~~(\star)$$
Also for a single parameter $\theta_i$ we know
$$\textbf{V}(\theta_i) \ge \frac{1}{\textbf{F}_{ii}} ~~~~~~~(\star \star)$$
But from $(\star)$ $$\textbf{C}(\theta_i) \ge (\textbf{F}^{-1})_{ii} ~~~~~~~(\star \star \star)$$
Given that from linear algebra $$(\textbf{F}^{-1})_{ii} \ge \frac{1}{\textbf{F}_{ii}},$$
how can we reconcile $(\star \star)$ and $(\star \star \star)$?