NP-completeness of a problem with pretty fast algorithm
Yes, such problems can be NP-complete.

Consider classical NP-complete graph problems like clique.

Clique has an $O(2^n)$ time algorithm, where $n$ is the number of vertices. However the input for clique is the adjacency matrix of the graph, which has $n^2$ bits. Therefore when the size of input is measured in bits, clique has an $O(2^\sqrt{n})$ time algorithm, which is $O(2^{\alpha n})$ for any positive $\alpha$.