Just answered my own question through experimenting. The following relates to a GridView control so the code behind is a bit more complex than would be the typical case. The code behind that I used for a simple ASP.NET Table control is listed toward the bottom of this post below the GridView code behind example.
aspx markup (two text boxes):
Box #1:
<EditItemTemplate>
<CE:Editor ID="CeName" runat="server" AutoConfigure="None" Text='<%# Eval("Name") %>'>
</CE:Editor>
</EditItemTemplate>
Box #2:
<EditItemTemplate>
<CE:Editor ID="CeDescription" runat="server" AutoConfigure="Minimal" Text='<%# Eval("Description") %>'>
</CE:Editor>
</EditItemTemplate>
Box #1:
string
name = ((Editor)grid.Rows[e.RowIndex].FindControl("CeName")).Text;
Box #2:
string
description = ((Editor)grid.Rows[e.RowIndex].FindControl("CeDescription")).Text;
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Here is the code behind for the editor when used with an ASP.NET Table control (the aspx markup is the same except that it isn't enclosed in a template):
// Get the values from the Cute Editor
string newName = CeNewName.Text; // CeNewName is the ID for the particulat text box
string newDescription = CeNewDescription.Text;