In this work, we focus on several completion problems for subclasses of chordal graphs: minimum Fill-In, Interval Completion, Proper Interval Completion, Threshold Completion, and Trivially Perfect Completion. In these problems, the task is to add at most k edges to a given graph in order to obtain a chordal, interval, proper interval, threshold, or trivially perfect graph, respectively. We prove the following lower bounds for all these problems, as well as for the related Chain Completion problem: Assuming the Exponential Time Hypothesis, none of these problems can be solved in time 2σ(n^1/2/log^cn) or 2σ (k^l/4/log^ck). nσ (1), for some integer c. Assuming the non-existence of a subexponential-time approximation scheme for Min Bisection on d-regular graphs, for some constant d, none of these problems can be solved in time 2°(n) or 2° (√nk). nσ(1). The second result is an evidence, that a significant improvement of the best known algorithms for parameterized completion problems would lead to a surprising breakthrough in the design of approximation algorithms for MlN bisection.