Sparse linear regression is the well-studied inference problem where one is given a design matrix $\mathbf{A} \in \mathbb{R}^{M\times N}$ and a response vector $\mathbf{b} \in \mathbb{R}^M$, and the goal is to find a solution $\mathbf{x} \in \mathbb{R}^{N}$ which is $k$-sparse (that is, it has at most $k$ non-zero coordinates) and minimizes the prediction error $\|\mathbf{A} \mathbf{x} - \mathbf{b}\|_2$. On the one hand, the problem is known to be $\mathcal{NP}$-hard which tells us that no polynomial-time algorithm exists unless $\mathcal{P} = \mathcal{NP}$. On the other hand, the best known algorithms for the problem do a brute-force search among $N^k$ possibilities. In this work, we show that there are no better-than-brute-force algorithms, assuming any one of a variety of popular conjectures including the weighted $k$-clique conjecture from the area of fine-grained complexity, or the hardness of the closest vector problem from the geometry of numbers. We also show the impossibility of better-than-brute-force algorithms when the prediction error is measured in other $\ell_p$ norms, assuming the strong exponential-time hypothesis.