Let G be a simple graph and It(G) denote the t-path ideal of G. It is well known that the Castelnuovo–Mumford regularity reg(R/It(G)) and the projective dimension pd(R/It(G)) are bounded below by the quantities (t-1)νt(G) and the big height bight(It(G)), respectively, where νt(G) denotes induced matching number of the hypergraph corresponding to It(G). We show that if t≥4, then the difference between reg(R/It(G)) and (t-1)νt(G), and the difference between pd(R/It(G)) and bight(It(G)) can be arbitrarily large even if we take G to be a tree. This, in particular, disproves a conjecture in Hang and Vu (Graphs Combin 41(1):18, 2025). However, when t=3 and G is chordal, we show that reg(R/I3(G))=2ν3(G) and pd(R/I3(G))=bight(I3(G)), extending the well-known formulas for the edge ideals of chordal graphs. As a consequence, we get that the 3-path ideal of a chordal graph is Cohen–Macaulay if and only if it is unmixed. Additionally, we show that the Alexander dual of I3(G) is vertex splittable when G is a tree, thereby resolving the t=3 case of a recent conjecture in Abdelmalek et al. (Int J Algebra Comput 33(3):481–498, 2023). Also, for each t≥3, we give examples of chordal graphs G such that the duals of the corresponding t-path ideals are not vertex splittable. Furthermore, we extend the formula of the regularity of 3-path ideals of chordal graphs to all t-path ideals of caterpillar graphs.