Various "black holes" are simply solutions to the Einstein Field Equations, and, if the EFE are an accurate picture of reality, then "bent time" (nontrivial spacetime curvature tensor) is exactly what the Einstein Field Equations tell us. In particular, the EFE predict a situation where, if there is enough energy in a region of space, then the geometry of spacetime will be such that all the future local lightcones of points within that region will be contained within the boundary (the Event Horizon) of that region, so that escape from that region would be tantamount to travelling backwards in time.
So, if we believe the Einstein Field Equations are accurate, the we are forced logically to the conclusion of "bent spacetime".
The question then becomes one of what grounds we have for believing the Einstein Field Equations. So far, almost all the predictions of the theory that we can think of have been experimentally verified: correct apsidal precession of planetary orbits (as done with Mercury), gravitational lensing, gravitational redshift, Shapiro delay, gravitational waves (spectacularly detected twice so far last year by LIGO) and the various precessional effects (de Sitter precession and Lense-Thirring frame dragging) of the meridian plane of orbits by Gravity Probe B, LAGEOS and others). The "Tests of General Relativity" Wikipedia Page has an extensive summary.
It's worth stating the spectacular agreement between the theoretically foretold and experimentally observed shape of the gravitational wave evolution with time for two inspiralling black holes that was found in both gravitational wave LIGO detections. That tells us that the EFE are a good description even in some pretty strong gravitational situations, although I don't think many physicists would believe the EFE are valid at all gravitational strengths. But they have been verified for systems of sufficient gravitational strength to severely "bend time" in any reasonable interpretation of your question.
 Such horizons always contain gravitational singularities, where the EFEs have an infinite divergence in all co-ordinate systems (so we believe they become invalid there) but it has not been proven that the converse is true (that every singularity has an event horizon). The converse is conjectured to be true (Penrose's "Cosmic Censorship Conjecture") but it is true that a naïve use of the EFEs can lead to "naked singularities" without horizons (e.g.
the Kerr metric for a spinning black hole with high enough angular momentum or the Reissner-Nördstrom metric for a charged black hole if the electric charge is big enough). However, for example, if one tried to assemble a Reissner-Nördstrom black hole with naked singularity by crushing electric charge together, the equations are such that the spacetime curvature effects end up growing with the increasing electrical potential energy faster than the charge can "shrink" the horizon such that no such attempt can succeed. Penrose's conjecture is that like effects prevent the assembling of a naked singularity in Nature.