We study the behavior of the posterior distribution in high-dimensional Bayesian Gaussian linear regression models having $p\gg n$, with $p$ the number of predictors and $n$ the sample size. Our focus is on obtaining quantitative finite sample bounds ensuring sufficient posterior probability assigned in neighborhoods of the true regression coefficient vector, $β^0$, with high probability. We assume that $β^0$ is approximately $S$-sparse and obtain universal bounds, which provide insight into the role of the prior in controlling concentration of the posterior. Based on these finite sample bounds, we examine the implied asymptotic contraction rates for several examples showing that sparsely-structured and heavy-tail shrinkage priors exhibit rapid contraction rates. We also demonstrate that a stronger result holds for the Uniform-Gaussian\footnote[2]{A binary vector of indicators ($γ$) is drawn from the uniform distribution on the set of binary sequences with exactly $S$ ones, and then each $β_i\sim\mathcal{N}(0,V^2)$ if $γ_i=1$ and $β_i=0$ if $γ_i=0$.} prior. These types of finite sample bounds provide guidelines for designing and evaluating priors for high-dimensional problems.