Short-run price competition with capacity contraints
If Firm 1 assumes Firm 2 produces and sells the maximum it can produce, i.e. $\hat{q}_2$, then Firm 1 can aim to maximise its revenue and profit $$\Pi_1=q_1p = q_1(1-q_1-\hat{q}_2)$$ which is maximised at $\left(\dfrac{1-\hat{q}_2}{2}\right)^2$ with $q_1=\dfrac{1-\hat{q}_2}{2}$, at least when Firm 1's maximum production exceeds this quantity. If not, then Firm 1 should continue to produce as much as possible.
So if Firm 1 is big enough then both firms producing as much as possible may not be an equilibrium

As an example, suppose $\hat{q}_1=0.5$ and $\hat{q}_2=0.1$, with a sum of $0.6=\frac35$ as required. If both firms produce at maximum capacity then the price will be $1-0.5-0.1=0.4$ making the Firm 1's revenue and profit $0.5\times 0.4=0.2$ 

But if Firm 2 produces at maximum capacity and Firm 1 were to produce only $0.45$, this would make the price $1-0.45-0.1=0.45$ and the Firm 1 's revenue and profit $0.45\times 0.45=0.2025$ which is higher than $0.2$, and the Firm 2 will also see profits rise due to the price increase, showing that both firms being at maximum production would not be an equilibrium in this case  

With the overall maximum production being $\frac35=0.6$, the equilibrium will break down in this way if $\hat{q}_1\gt \frac25 =0.4$ and $\hat{q}_2 \lt \frac15 =0.2$