C# Inheritance – Why List Cannot Be Assigned to List

No - don't use new here; that doesn't give you polymorphism. List<T> isn't intended for inheritance in this way; use Collection<T> and override the Add InsertItem method.

public class LimitedCollection<T> : Collection<T>
{
    private int _maxitems = 500;

    protected override void InsertItem(int index, T item)
    {
        base.InsertItem(index, item);
        TrimData(); /* Delete old data if lenght too long */
    }

    private void TrimData()
    {
        int num = Math.Max(0, base.Count - _maxitems);
        while (num > 0)
        {
            base.RemoveAt(0);
            num--;
        }
    }
}

There are some good answers here. I would add to them the following points.

What is the correct C# way of representing a data structure, which, "logically" (that is to say, "to the human mind") is just a list of things with a few bells and whistles?

Ask any ten non-computer-programmer people who are familiar with the existence of football to fill in the blank:

A football team is a particular kind of _____

Did anyone say "list of football players with a few bells and whistles", or did they all say "sports team" or "club" or "organization"? Your notion that a football team is a particular kind of list of players is in your human mind and your human mind alone.

List<T> is a mechanism. Football team is a business object -- that is, an object that represents some concept that is in the business domain of the program. Don't mix those! A football team is a kind of team; it has a roster, a roster is a list of players. A roster is not a particular kind of list of players. A roster is a list of players. So make a property called Roster that is a List<Player>. And make it ReadOnlyList<Player> while you're at it, unless you believe that everyone who knows about a football team gets to delete players from the roster.

Is inheriting from List<T> always unacceptable?

Unacceptable to whom? Me? No.

When is it acceptable?

When you're building a mechanism that extends the List<T> mechanism.

What must a programmer consider, when deciding whether to inherit from List<T> or not?

Am I building a mechanism or a business object?

But that's a lot of code! What do I get for all that work?

You spent more time typing up your question that it would have taken you to write forwarding methods for the relevant members of List<T> fifty times over. You're clearly not afraid of verbosity, and we are talking about a very small amount of code here; this is a few minutes work.

UPDATE

I gave it some more thought and there is another reason to not model a football team as a list of players. In fact it might be a bad idea to model a football team as having a list of players too. The problem with a team as/having a list of players is that what you've got is a snapshot of the team at a moment in time. I don't know what your business case is for this class, but if I had a class that represented a football team I would want to ask it questions like "how many Seahawks players missed games due to injury between 2003 and 2013?" or "What Denver player who previously played for another team had the largest year-over-year increase in yards ran?" or "Did the Piggers go all the way this year?"

That is, a football team seems to me to be well modeled as a collection of historical facts such as when a player was recruited, injured, retired, etc. Obviously the current player roster is an important fact that should probably be front-and-center, but there may be other interesting things you want to do with this object that require a more historical perspective.