I have recently encountered this idea of "negative activation energy" as I was reading Chapter 11 of Elements of Physical Chemistry (5th Edition). On p. 252, the kinetics of the following reaction was studied:
$$\ce {2 NO (g) + O2 (g) -> 2 NO2 (g)}$$
A plausible and simple mechanism which was selected for this illustration, involved two steps: Firstly, the formation of the $\ce {N2O2}$ dimer from a collision of the two $\ce {NO}$ reactant molecules. It is then followed by collision of this dimer with an $\ce {O2}$ molecule. By applying the steady-state approximation (i.e. the rate of change of the dimer with respect to time approximately equals $\ce {0}$), the following overall rate constant is obtained:
$$k_\mathrm{r} = \frac{k_1 k_2}{k_1'}$$
$k_\mathrm{r}$ is the overall rate constant, $k_1$ is the forward rate constant for step $1$, $k_1'$ is the reverse rate constant for step $1$ and $k_2$ is the forward rate constant for step $2$.
When temperature increases, all of these individual rate constants for these elementary reactions increase. However, the extent of increase in their values differs between these individual rate constants. For this particular reaction, Atkins & Paula (2009) highlight that as temperature increases, the overall rate constant actually decreases. This is because the rate constant for the dissociation of the dimer increases much more as temperature increases, compared to the other rate constants in the numerator.
Mathematically, if we were to use the Arrhenius relationship for the rate constant for the overall reaction, we would say that the reaction has "negative activation energy" since that is the only way to have the rate constant decreasing in value as temperature increases (as $R, T > 0$), but it is pertinent to keep in mind that the rate of each elementary step does in fact increase as temperature increases. However, the extent to which they individually increase differs between each elementary step.
This is perhaps one way to see how a reaction can be said to have "negative activation energy" overall.
Reference
Atkins, P. W.; Paula, J. Elements of Physical Chemistry (5th ed.). W. H. Freeman & Company, 2009.