As pdnnood stated only if you save as a .pdn paint net file but even then all you can do is move or rescale the text, to write something new you will have to deselect that layer and create a new one with the updated text on it.
for example:
You open up your map into pdn and it becomes the background layer
you press add new layer and on this you write England with an arrow pointing to England
You then save this as a .pdn file then save again as a jpeg or png
you then shut down your pc and leave it for a day
next day you start up your pc and look at your jpeg image,you notice that it points to France by mistake ( you can do nothing about it in its jpeg state )
you load in the .pdn file and go to layer 1 ( text layer ) you select the blue arrow ( move selection ) click on the layer and move the text to the correct place,problem solved.
While moving the text you realise that you spelt England wrong ( nothing you can do about this ) apart from turning layer off and adding a new layer and re-writing the text again correctly.