If you haven't already, try checking these two items under the "object" menu when your embedded (anchored) text box is direct-selected, or selcted with the cursor as if it were text:
1. Object > Anchored Object > Options, then check that the settings don't contain space before or space after. How the other options are set may affect how your text block positions at the edge of a box, for instance at the top of the dialog box Position: Inline/above Line is the default, but some custom setting may be chosen. If so, there are quite a few additional settings in the "custom" configuration.
2. Object > Text Frame Options > Baseline Options "tab", check if Offset is changed to "Ascent" or "Leading" or any other choice, if that changes the position of your embedded text box (have the preview box checked at the bottom left of the dialog box so you can see how the change affects your anchored box)... For this setting to affect your anchored box, you need to select it as if it were a paragraph. So, "highlight" it with the text cursor, rather than direct selecting it with the pointer... because the Text Frame options setting is really directed at the enclosing text box, not the embedded text box as an object.
Any paragraph or character formatting you have going on in the enclosing text box at the line where the anchored text box is placed (attached to) can adversely affect your anchored box's position. So things like leading, space before/space after, lock to baseline, baseline shift, etc. that is present at the insertion point where you copied or inserted your inside text box, can affect its position, even if there is no type present to show the style or attribute.
Hope this helps.