I have simple code that creates connection to Firebird (any version, from 2.5 to 5.0).

C#, x64, FbConnection, 1000 connections, all working fine:

List<FbConnection> fbList = new List<FbConnection>();
for (int i = 1; i <= 1000; i++)
{
    try
    {

        var connection = new FbConnection("User=SYSDBA;Password=masterkey;Database=localhost/3053:c:\rollcontrol\rollcontrol;Pooling=false; MinPoolSize=0;MaxPoolSize=2000");
        connection.Open();

        fbList.Add(connection);
        Console.WriteLine(i.ToString());

    }
    catch (Exception e)
    {
        Console.WriteLine(DateTime.Now.ToString() + ": " + e.ToString());
    }
}

C#, x64, OdbcConnection, 1000 connections, process stopping on 180 connections (no error, the process just hangs):

List<OdbcConnection> odbcList = new List<OdbcConnection>();
for (int i = 1; i <= 1000; i++)
{
    try {

        OdbcConnection DbConnection = new OdbcConnection("DSN=rollcontrol");
        DbConnection.Open();

        odbcList.Add(DbConnection);
        Console.WriteLine(i.ToString());

    } catch (Exception e) { 
        Console.WriteLine(DateTime.Now.ToString() + ": " + e.ToString());
    }
}

ODBC has a limit on the number of connections? Maybe, it is possible to increase the limit number of connections?

UPDATE

Problem was in ODBC driver. Version 2.05 has a limitation, version 3.0 working fine