¡Este error me vuelve loco y ni siquiera puedo reproducirlo!

He publicado mi sitio web para Windows Azure y SQL Azure y funcionó muy bien. De repente, y no funciona, me sale este desagradable error:

Aquí está el error que recibo:

http://jsfiddle.net/shimmy/pcS7g/embedded/result

Aquí está el error:

Se produjo un error relacionado con la red o específico de la instancia al establecer una conexión a SQL Server. El servidor no se encontró o no estaba accesible. Verifique que el nombre de la instancia sea correcto y que SQL Server esté configurado para permitir conexiones remotas. (proveedor: interfaces de red SQL, error: 26 - Error al localizar el servidor / Especificación especificada)

Quiero enfatizar que puedo acceder al SQL a través de SSMS o VS y ver mis tablas, etc.

No tengo idea de qué causa este error molesto que ocurre una y otra vez después de cada implementación.

Rastreo de pila:

[SqlException (0x80131904): A network-related or instance-specific error occurred while establishing a connection to SQL Server. The server was not found or was not accessible. Verify that the instance name is correct and that SQL Server is configured to allow remote connections. (provider: SQL Network Interfaces, error: 26 - Error Locating Server/Instance Specified)]
   System.Data.SqlClient.SqlInternalConnection.OnError(SqlException exception, Boolean breakConnection, Action`1 wrapCloseInAction) +5295167
   System.Data.SqlClient.TdsParser.ThrowExceptionAndWarning(TdsParserStateObject stateObj, Boolean callerHasConnectionLock, Boolean asyncClose) +242
   System.Data.SqlClient.TdsParser.Connect(ServerInfo serverInfo, SqlInternalConnectionTds connHandler, Boolean ignoreSniOpenTimeout, Int64 timerExpire, Boolean encrypt, Boolean trustServerCert, Boolean integratedSecurity, Boolean withFailover) +5307115
   System.Data.SqlClient.SqlInternalConnectionTds.AttemptOneLogin(ServerInfo serverInfo, String newPassword, SecureString newSecurePassword, Boolean ignoreSniOpenTimeout, TimeoutTimer timeout, Boolean withFailover) +145
   System.Data.SqlClient.SqlInternalConnectionTds.LoginNoFailover(ServerInfo serverInfo, String newPassword, SecureString newSecurePassword, Boolean redirectedUserInstance, SqlConnectionString connectionOptions, SqlCredential credential, TimeoutTimer timeout) +920
   System.Data.SqlClient.SqlInternalConnectionTds.OpenLoginEnlist(TimeoutTimer timeout, SqlConnectionString connectionOptions, SqlCredential credential, String newPassword, SecureString newSecurePassword, Boolean redirectedUserInstance) +307
   System.Data.SqlClient.SqlInternalConnectionTds..ctor(DbConnectionPoolIdentity identity, SqlConnectionString connectionOptions, SqlCredential credential, Object providerInfo, String newPassword, SecureString newSecurePassword, Boolean redirectedUserInstance, SqlConnectionString userConnectionOptions) +434
   System.Data.SqlClient.SqlConnectionFactory.CreateConnection(DbConnectionOptions options, DbConnectionPoolKey poolKey, Object poolGroupProviderInfo, DbConnectionPool pool, DbConnection owningConnection, DbConnectionOptions userOptions) +5309659
   System.Data.ProviderBase.DbConnectionFactory.CreateNonPooledConnection(DbConnection owningConnection, DbConnectionPoolGroup poolGroup, DbConnectionOptions userOptions) +38
   System.Data.ProviderBase.DbConnectionFactory.TryGetConnection(DbConnection owningConnection, TaskCompletionSource`1 retry, DbConnectionOptions userOptions, DbConnectionInternal& connection) +5311874
   System.Data.ProviderBase.DbConnectionClosed.TryOpenConnection(DbConnection outerConnection, DbConnectionFactory connectionFactory, TaskCompletionSource`1 retry, DbConnectionOptions userOptions) +143
   System.Data.SqlClient.SqlConnection.TryOpen(TaskCompletionSource`1 retry) +83
   System.Data.SqlClient.SqlConnection.Open() +96
   System.Web.Management.SqlServices.GetSqlConnection(String server, String user, String password, Boolean trusted, String connectionString) +76

[HttpException (0x80004005): Unable to connect to SQL Server database.]
   System.Web.Management.SqlServices.GetSqlConnection(String server, String user, String password, Boolean trusted, String connectionString) +131
   System.Web.Management.SqlServices.SetupApplicationServices(String server, String user, String password, Boolean trusted, String connectionString, String database, String dbFileName, SqlFeatures features, Boolean install) +89
   System.Web.Management.SqlServices.Install(String database, String dbFileName, String connectionString) +27
   System.Web.DataAccess.SqlConnectionHelper.CreateMdfFile(String fullFileName, String dataDir, String connectionString) +386

Actualizar

Aquí hay una copia de las cadenas de conexión tal como aparecen en web.config del servidor:

<connectionStrings>
  <add name="Context" connectionString="Data Source=tcp:abcdefg.database.windows.net,1433;Initial Catalog=Database;User Id=shimmy@tcp:abcdefg;Password=my123password;" providerName="System.Data.SqlClient" />
  <add name="Storage" connectionString="DefaultEndpointsProtocol=https;AccountName=accountname;AccountKey=accountkey==" />
</connectionStrings>

Incluso intenté eliminar las cadenas de conexión por completo de web.config del servidor. No tengo ni idea de dónde estáApp_Data ¡La cosa viene de aquí!

MiDbContext Se ve como esto:

public Context()
  : base("name=Context") //I also tried "Context" alone
{
}