Change this to "the accusing company would get a court order for a take down notice".
The problem is that take-down notices coming from companies should have no legal weight. Even those coming from a company's lawyer should have no weight until a court/judge decides there is enough evidence to warrant one.
Getting a take down notice SHOULD require a procedure similar to a search warrant. A judge should have to sign off that a legal infraction has been made before the take-down notice is issued.
However, as a hosting provider, if a company sent me a notice that a client was using the site for illegal purposes, I would investigate, contact the client and, if I could not determine that an infringement was being done, have my lawyer contact the accusing company for more information and a request for a legal take-down notice, if that is what is required.
Unfortunately, with this precedent set, companies can issue take-down notices and file lawsuits against hosting providers with little proof that an infringement is taking place. Most hosting providers do not have deep enough pockets to fight a big lawsuit. Until a provider does fight back - this precedent will stand.