Yes, you can switch on the name...

switch (obj.GetType().Name)
{
case "TextBox":...
}

Update C# 7

Yes: Source

switch(shape)
{
case Circle c:
WriteLine($"circle with radius {c.Radius}");
break;
case Rectangle s when (s.Length == s.Height):
WriteLine($"{s.Length} x {s.Height} square");
break;
case Rectangle r:
WriteLine($"{r.Length} x {r.Height} rectangle");
break;
default:
WriteLine("<unknown shape>");
break;
case null:
throw new ArgumentNullException(nameof(shape));
}

Prior to C# 7

No.

http://blogs.msdn.com/b/peterhal/archive/2005/07/05/435760.aspx

We get a lot of requests for addditions to the C# language and today I'm going to talk about one of the more common ones - switch on type. Switch on type looks like a pretty useful and straightforward feature: Add a switch-like construct which switches on the type of the expression, rather than the value. This might look something like this:

switch typeof(e) {
case int:    ... break;
case string: ... break;
case double: ... break;
default:     ... break;
}

This kind of statement would be extremely useful for adding virtual method like dispatch over a disjoint type hierarchy, or over a type hierarchy containing types that you don't own. Seeing an example like this, you could easily conclude that the feature would be straightforward and useful. It might even get you thinking "Why don't those #*&%$ lazy C# language designers just make my life easier and add this simple, timesaving language feature?"

Unfortunately, like many 'simple' language features, type switch is not as simple as it first appears. The troubles start when you look at a more significant, and no less important, example like this:

class C {}
interface I {}
class D : C, I {}


switch typeof(e) {
case C: … break;
case I: … break;
default: … break;
}

Link: https://blogs.msdn.microsoft.com/peterhal/2005/07/05/many-questions-switch-on-type/

The simplest thing to do could be to use dynamics, i.e. you define the simple methods like in Yuval Peled answer:

void Test(WebControl c)
{
...
}


void Test(ComboBox c)
{
...
}

Then you cannot call directly Test(obj), because overload resolution is done at compile time. You have to assign your object to a dynamic and then call the Test method:

dynamic dynObj = obj;
Test(dynObj);

Here's an option that stays as true I could make it to the OP's requirement to be able to switch on type. If you squint hard enough it almost looks like a real switch statement.

The calling code looks like this:

var @switch = this.Switch(new []
{
this.Case<WebControl>(x => { /* WebControl code here */ }),
this.Case<TextBox>(x => { /* TextBox code here */ }),
this.Case<ComboBox>(x => { /* ComboBox code here */ }),
});


@switch(obj);

The x in each lambda above is strongly-typed. No casting required.

And to make this magic work you need these two methods:

private Action<object> Switch(params Func<object, Action>[] tests)
{
return o =>
{
var @case = tests
.Select(f => f(o))
.FirstOrDefault(a => a != null);


if (@case != null)
{
@case();
}
};
}


private Func<object, Action> Case<T>(Action<T> action)
{
return o => o is T ? (Action)(() => action((T)o)) : (Action)null;
}

Almost brings tears to your eyes, right?

Nonetheless, it works. Enjoy.

The following code works more or less as one would expect a type-switch that only looks at the actual type (e.g. what is returned by GetType()).

public static void TestTypeSwitch()
{
var ts = new TypeSwitch()
.Case((int x) => Console.WriteLine("int"))
.Case((bool x) => Console.WriteLine("bool"))
.Case((string x) => Console.WriteLine("string"));


ts.Switch(42);
ts.Switch(false);
ts.Switch("hello");
}

Here is the machinery required to make it work.

public class TypeSwitch
{
Dictionary<Type, Action<object>> matches = new Dictionary<Type, Action<object>>();
public TypeSwitch Case<T>(Action<T> action) { matches.Add(typeof(T), (x) => action((T)x)); return this; }
public void Switch(object x) { matches[x.GetType()](x); }
}