We study a linear cocycle over the irrational rotation $$\sigma_{\omega}(x)=x+\omega$$ of the circle $$\mathbb{T}^{1}$$. It is supposed that the cocycle is generated by a $$C^{2}$$-map$$A_{\varepsilon}:\mathbb{T}^{1}\to SL(2,\mathbb{R})$$ which depends on a small parameter $$\varepsilon\ll 1$$ and has the form of the Poincaré map corresponding to a singularly perturbed Hill equation with quasi-periodic potential. Under the assumption that the norm of the matrix $$A_{\varepsilon}(x)$$ is of order $$\exp(\pm\lambda(x)/\varepsilon)$$, where $$\lambda(x)$$ is a positive function, we examine the property of the cocycle to possess an exponential dichotomy (ED) with respect to the parameter $$\varepsilon$$. We show that in the limit $$\varepsilon\to 0$$ the cocycle “typically” exhibits ED only if it is exponentially close to a constant cocycle.Conversely, if the cocycle is not close to a constant one,it does not possess ED, whereas the Lyapunov exponent is “typically” large.