Here are a few things to consider...
You should consider what the original company's DBA (Doing Business As) is. More then likely, MyCompany Inc. has registered a DBA as MyCompany. You can have multiple companies using the same DBA, but they have to agree with each other.
Brand visibility is hard enough without having to deal with confusion as to what brand is what. How will your customers find you?
Devaluation to brand
Imagine if you found a Microsoft SP - you can probably tell fairly quickly who was the rip off, even if Microsoft SP doesn't sell related products.
Your brand is all you have to represent your company/product to your consumers. Almost any other name is better then a ripoff of someone else.