My name is Andrew, I am the Database Administrator for a business with 300+ staff in Invercargill, New Zealand.
Good news: this error message has *no* indication that the database itself is corrupt; anyone who claims this is wrong. There are two possible causes: your client software is not closing connections after querying the database, or your SQL Server installation is configured incorrectly.
If you chose me to resolve this issue, I would connect to your server via TeamViewer (or any remote access tool you prefer), and would then ask you to demonstrate the issue (if you can run the client on the server), and inventify and resolve the cause of this issue, allowing your client software to connect again.
If this took under three hours, and you would appreciate having a backup, I would then go about creating a script to backup your database on a routine basis (of your choice) to a password protected backup on your server.
If the cause of the connection limit reached error is a flaw in the client software's code, I would review possible solutions to this, and either request a job to modify the client software or recommend a solution.
I am available to discuss the issue with via Skype or email.
Thanks for reading my proposal,
Andrew