Se por "número de conexões" você quer dizer o tamanho do pool de conexão, isso se aplica:
A connection pool is created for each unique connection string. When a pool is created, multiple connection objects are created and added to the pool so that the minimum pool size requirement is satisfied. Connections are added to the pool as needed, up to the maximum pool size specified (100 is the default). Connections are released back into the pool when they are closed or disposed.
When a SqlConnection object is requested, it is obtained from the pool if a usable connection is available. To be usable, a connection must be unused, have a matching transaction context or be unassociated with any transaction context, and have a valid link to the server.
The connection pooler satisfies requests for connections by reallocating connections as they are released back into the pool. If the maximum pool size has been reached and no usable connection is available, the request is queued. The pooler then tries to reclaim any connections until the time-out is reached (the default is 15 seconds). If the pooler cannot satisfy the request before the connection times out, an exception is thrown.
Fonte: Pool de Conexões do SQL Server (ADO.NET)
O tamanho padrão dos pools de conexão é 100. Você pode alterar esse tamanho na própria connectionstring, mas você deve ter um bom motivo para fazer isso. Na maioria dos casos em que você fica sem conexões, elas estão vazando (= não fechadas corretamente) no aplicativo.