How can I solve a connection pool problem between ASP.NET and SQL Server?

Unless your usage went up a lot, it seems unlikely that there is just a backlog of work. IMO, the most likely option is that something is using connections and not releasing them promptly. Are you sure you are using using in all cases? Or (through whatever mechanism) releasing the connections?

In most cases connection pooling problems are related to connection leaks. Your application probably doesn't close its database connections correctly and consistently. When you leave connections open, they remain blocked until the .NET garbage collector closes them for you by calling their Finalize() method.

You want to make sure that you are really closing the connection. For example the following code will cause a connection leak, if the code between .Open and Close throws an exception:

var connection = new SqlConnection(connectionString);

connection.Open();
// some code
connection.Close();                

The correct way would be this:

var connection = new SqlConnection(ConnectionString);

try
{
     connection.Open();
     someCall (connection);
}
finally
{
     connection.Close();                
}

or

using (SqlConnection connection = new SqlConnection(connectionString))
{
     connection.Open();
     someCall(connection);
}

When your function returns a connection from a class method make sure you cache it locally and call its Close method. You'll leak a connection using this code for example:

var command = new OleDbCommand(someUpdateQuery, getConnection());

result = command.ExecuteNonQuery();
connection().Close(); 

The connection returned from the first call to getConnection() is not being closed. Instead of closing your connection, this line creates a new one and tries to close it.

If you use SqlDataReader or a OleDbDataReader, close them. Even though closing the connection itself seems to do the trick, put in the extra effort to close your data reader objects explicitly when you use them.

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This article "Why Does a Connection Pool Overflow?" from MSDN/SQL Magazine explains a lot of details and suggests some debugging strategies:

• Run sp_who or sp_who2. These system stored procedures return information from the sysprocesses system table that shows the status of and information about all working processes. Generally, you'll see one server process ID (SPID) per connection. If you named your connection by using the Application Name argument in the connection string, your working connections will be easy to find.
• Use SQL Server Profiler with the SQLProfiler TSQL_Replay template to trace open connections. If you're familiar with Profiler, this method is easier than polling by using sp_who.
• Use the Performance Monitor to monitor the pools and connections. I discuss this method in a moment.
• Monitor performance counters in code. You can monitor the health of your connection pool and the number of established connections by using routines to extract the counters or by using the new .NET PerformanceCounter controls.

Did you check for DataReaders that are not closed and response.redirects before closing the connection or a datareader. Connections stay open when you dont close them before a redirect.

Upon installing .NET Framework v4.6.1 our connections to a remote database immediately started timing out due to this change.

To fix simply add the parameter TransparentNetworkIPResolution in the connection string and set it to false:

Server=myServerName;Database=myDataBase;Trusted_Connection=True;TransparentNetworkIPResolution=False