Yes, it totally can happen. I've seen money be saved by doing it.
This is not a tech decision, it's a business decision. Code rewrites are long term gains, while "if it ain't totally broke..." is a short term gain. If you are in a first year startup that is focused on getting a product out the door, the answer is usually to just live with it. If you're in an established company, or the errors with the current systems are causing more workload, therefor more company money.. then they might go for it.
Present the problem as best as you can to your GM, use dollar values where you can. "I don't like dealing with it" means nothing. "It'll take twice the time to do everything until this is fixed" means a lot.
