CL350 pilot reported receiving a low altitude warning from ATC while being vectored for approach. Flight crew climbed to a safe altitude and continued approach.

Narrative

On Day 0; I was acting pilot-in-command; and pilot flying; of a Challenger 350 from ZZZ to ZZZ1 on an IFR flight plan. At Approximately XA:45; while being vectored for the ILS RWY XX into ZZZ1; we were advised by TRACON to 'check attitude' and 'climb to 4000 feet.' We immediately complied with a climb from approximately 2250 feet to 4000 feet. We were then given vectors and lower altitudes and cleared for the ILS RWY XX approach in to ZZZ1. The remainder of the flight was without incident.After landing; we were given a request to contact the tower; which we complied with. The supervisor at the time; Person A; collected my personal information and advised us he would listen to the 'tapes' and look at the data and may send this to the FSDO. He didn't know at the time if this was a pilot deviation or not. I have called the tower numerous times to obtain more information; but they have been too busy to talk about it.Retracing what led up to this is as follows: Upon handoff to ZZZ TRACON; three attempts were made to contact them before an acknowledgement was received. This was; in my opinion; due to the controller being saturated with additional aircraft transitioning through TRACON's airspace due to storms and buildups. The controllers initial response was curt and sounded annoyed. Earlier in the flight we had momentary trouble communicating with ZZZ Center; which I believe was due to the electrical storms; and may have contributed to requiring three attempts to contact ZZZ TRACON? After checking in with TRACON; we were given multiple lower altitudes and headings and eventually given direct to ZZZZZ; an IF/IAF fix for the ILS XX and an altitude of 4000 feet and then eventually 3000 feet. The altitudes were not given with adequate distance out to allow us to be at an altitude we could join the approach for a landing. Another company aircraft arriving from the west were not given adequate descent clearances as well; causing for pilot requests for lower altitudes and then in our case; being taken through the final approach course for the ILS RWY XX. The controller opted then to bring in the other aircraft first and gave us a heading of 320 and a 'descend to 4000 feet.' My co-pilot put 2000 feet in our altitude selector and I bugged us to a heading of 320. I didn't catch/see he lowered the altitude to 2000 feet rather than 4000 feet. After listening to the transcripts of liveATC; my co-pilot only replies with 'vectors Aircraft X.' ATC did not query us to verify we received the correct heading and altitude. I don't know if we had radio reception/interference again or my SIC was saturated with additional changes to our routing and altitudes along with reloading the FMS and running checklists? Or both? I don't recall specifically hearing '2000 feet' stated. Our company policy is for the pilot who adjust the altitude selector to leave their hand on it until the other acknowledges the correct change. My contract SIC did not do that. What took place is very concerning. In my XX+ years of flying I have never had a possible altitude deviation. My co-pilot and I debriefed and talked about this at length. One of the discussion points was to keep his hand on the altitude selector until the other has a chance to verify. He is a former military helicopter pilot with many years of experience in a variety of challenging and demanding environments. He is a professional and also rattled by this! As mentioned earlier; he was acting in capacity as a contract pilot and therefore the synergies of flying with one of my company co-workers was missing and believe that may have been a contributing factor. In addition; the weather and possible radio interference and TRACON saturation were contributing factors I believe as well. I have tried to piece this together with the best of my recollection and along with listening to the ATC tape transcript from liveATC. I have flown into ZZZ1 over XXX times and feel I know the area and system well anddesire to have a thorough understanding so this never happens again. Hopefully ATC will take my call to help me piece this together so I can learn from it and make necessary changes. Thank You!