Apparantley all aircraft recievers should have sufficient bandwidth to cope with recieving offset transmissions...that said I remember in my military days we could not talk to USAF F111 swing wing bombers if using multi channel TX/RX we had to select a single channel before their radios would recieve our transmissions. It apppears there are some certain types of radios on the market that have a poor band width capability..it would be interesting to know what radio you use and see if it is on the known list of offending radios.
Rather than "poor bandwidth capability" I think it's more likely to be due to the radio having good filtering abilities to cut out transmissions from other frequencies near to your operating frequency. It probably "thinks" that the offset transmissions are rogue and tries to cut them out rather than letting them through clearly on your current frequency.
I have the same problem with the scanner I use to listen to ATIS/ATC before I leave for the airfield. On certain frequencies the controllers cut out/sound distorted unless you detune the scanner slightly (something you can't do on an aircraft comms set
AFAIK)
Hope that made some sense
TrafficPilot