I don't have time to find it now, but I remember reading federal regulations state that money paid to the government must be deposited into an account held by the treasury within three business days of receipt of the money. The questionable part is when does the government take ownership of the money, when they receive the envelope in the mail or when they open and process the envelope?
To give you an example in regards to cashing the check, but not filling out the form correctly: When my wife applied for her first US Passport she went to an acceptance agent. The agent was happy to see that she had everything filled neatly since it was done on a computer and printed and all the necessary documents and a cashier's check ready that she forget to ask my wife to sign her passport application. The application was forwarded on to Passport Services a couple of days later, the cashier's check was cashed about a week after filling with the agent, and then nothing for six weeks. Six weeks after applying and several phone calls to the National Passport Information Center (NPIC) my wife received a letter in the mail from the local Passport Agency (DC) stating that she didn't sign her application. The letter directed her to go back to the agent and refile (no charge). My wife was so upset she called NPIC and said that she didn't have time to go to the agent and needed her passport within three weeks. NPIC informed her that she could go to the DC Passport Agency the next day without an appointment to take care of the issue. When she went to the DC Passport Agency the next day, the clerk got my wife's application when she was there and showed her that it wasn't signed, and had her sign it on the spot. My wife received her new US Passport in the mail three days later.