It is possible that you could be sued, yes. There is certainly no guarantee that you wouldn't be. Whether any trademark infringement lawsuit, filed against your Stewed Apples and Beyond by the existing Stewed Apples Inc, would be a successful one for Stewed Apples depends on a whole host of different factors, however.
Trademark law, by the way, protects the words, phrases, logos, symbols, shapes, colors, sounds, letters, numbers and anything else that is used to identify and distinguish one company and its products or services from another.
Copyright law protects original and creative expressions, like artwork, literary works, musical works, graphics, photographs, architecture and so on.
In some cases, there may be an overlap of copyright and trademark rights. This, for example, can be the case with logo designs. The original and creative design of the logo can be copyrighted and also serve as a company identifier (trademark).
It is always safer and wiser to choose a business name that is unlike any others being used commercially. When you have a name that is the same or similar to one already in existence, one that identifies an existing company and its products or services, you are at a greater risk of infringing on the other company's rights.
That is not to say that companies cannot share the same or similar names - they can and they do. It just means that the risk of a lawsuit is far greater when your company name is not unique to you alone.
If you are starting a company, I advise that you consult with an attorney. A review of your business plans, prior to spending a lot of time, money and effort implementing them, can save you from having legal headaches in the future.
Good luck.