Brandie - are you using SQL Aliases OR DNS Aliases? If it is DNS aliases, you will likely need to include the instance name (for a named instance) when you have the "MyInstanceName" in there. If it is a default instance, then just the server name should work.
If it is a SQL Alias, I believe you would need to push that alias out to all client machines for it to be recognized and the SQL Alias should include the port.
And to add to what Johan said, if port 1434 is blocked OR the SQL Browser Service isn't running, you will need to provide the port number when connecting as the client has no idea what port to connect to and there are a lot of them (65,535 if memory serves).
As for us making assumptions about your knowledge of SQL ports, we have to. You provided us no indication about how much you knew about SQL ports, so we have to assume something. And, in my opinion, it is MUCH safer to assume that the person knows nothing rather than assume they are an expert; especially when it has been pointed out that connecting including the port number worked, but was untested. I am sure that Johan meant no disrespect by what they said, he was just trying to help.
Now, to add to the above, I have used SQL Aliases a lot at my company (and been bitten a few times by them too) and have never needed to include the port nor have it running on port 1433. We have multiple named instances installed on a single box (don't ask why...) and our SQL aliases work without a problem. We looked into the DNS approach for aliases but decided against it as DNS replication to all of our sites doesn't happen immediately and in the event of a failover or an upgrade resulting in a new instance being created and things being migrated over, it is much easier to ask people to reboot and the logon script updates things than it is to get DNS updated and wait for it to replicate across all the DC's and out to all client workstations.
Just my 2 cents. I am glad you got it figured out, but I am not certain you are correct on it needing to use port 1433 for that to work.
The above is all just my opinion on what you should do.
As with all advice you find on a random internet forum - you shouldn't blindly follow it. Always test on a test server to see if there is negative side effects before making changes to live!