Here's my test code, which seems to suggest that it's better to connect multiple times instead of connecting just once.
Am I doing something wrong?
int numIts = 100;
Stopwatch sw = new Stopwatch();
sw.Start();
using (SqlConnection connection = new SqlConnection(connectionParameters))
{   
            connection.Open();
    for(int i = 0; i < numIts; i++)
    {
        SqlCommand command = new SqlCommand(sqlCommandName, connection);
                command.CommandType = CommandType.StoredProcedure;
                command.Parameters.AddWithValue(par1Name, par1Val);
                command.Parameters.AddWithValue(par2Name, par2Val);
        using(SqlDataReader reader = command.ExecuteReader())
        {
        }
    }
}
sw.Stop();
TimeSpan durationOfOneConnectionManyCommands = sw.Elapsed;
Console.WriteLine(durationOfOneConnectionManyCommands);
sw.Reset();
sw.Start();
for(int i = 0; i < numIts; i++)
{
    using (SqlConnection connection = new SqlConnection(connectionParameters))
    {   
                connection.Open();
        SqlCommand command = new SqlCommand(sqlCommandName, connection);
                command.CommandType = CommandType.StoredProcedure;
                command.Parameters.AddWithValue(par1Name, par1Val);
                command.Parameters.AddWithValue(par2Name, par2Val);
        using(SqlDataReader reader = command.ExecuteReader())
        {
        }
    }                               
}
sw.Stop();
TimeSpan durationOfManyConnections = sw.Elapsed;
Console.WriteLine(durationOfManyConnections);
Output:
//output:
//00:00:24.3898218   // only one connection established
//00:00:23.4585797   // many connections established.
//
//output after varying parameters (expected much shorter):
//00:00:03.8995448
//00:00:03.4539567
Update:
OK, so those who said it would be faster w/ one connection have it. (although the difference is marginal, if any.) Here's the revised code and output:
public void TimingTest()
{
    numIts = 1000;
    commandTxt = "select " + colNames + " from " + tableName;
    OneConnection();
    ManyConnections();
    OneConnection();
}
private void ManyConnections()
{
    Stopwatch sw = new Stopwatch();
    sw.Start();
    for (int i = 0; i < numIts; i++)
    {
        using (SqlConnection connection = new SqlConnection(connectionParameters))
        {
            connection.Open();
            using (SqlCommand command = connection.CreateCommand())
            {
                command.CommandText = commandTxt;
                using (SqlDataReader reader = command.ExecuteReader())
                {
                }
            }
        }
    }
    sw.Stop();
    TimeSpan durationOfManyConnections = sw.Elapsed;
    Console.WriteLine("many connections: " + durationOfManyConnections);
}
private void OneConnection()
{
    Stopwatch sw = new Stopwatch();
    sw.Start();
    using (SqlConnection connection = new SqlConnection(connectionParameters))
    {
        connection.Open();
        for (int i = 0; i < numIts; i++)
        {
            using (SqlCommand command = connection.CreateCommand())
            {
                command.CommandText = commandTxt;
                using (SqlDataReader reader = command.ExecuteReader())
                {
                }
            }
        }
    }
    sw.Stop();
    TimeSpan durationOfOneConnectionManyCommands = sw.Elapsed;
    Console.WriteLine("one connection: " + durationOfOneConnectionManyCommands);
}
Output:
one connection: 00:00:08.0410024
many connections: 00:00:08.7278090
one connection: 00:00:08.6368853
one connection: 00:00:10.7965324
many connections: 00:00:10.8674326
one connection: 00:00:08.6346272
Update:
the difference is more striking if I use SQLConnection.ClearAllPools() after each function:
Output:
one connection: 00:00:09.8544728
many connections: 00:00:11.4967753
one connection: 00:00:09.7775865
 
     
     
     
     
     
    