I'm an editor in the UK and yes, it is done. It depends, of course, on the relationship between the editor and the client. I have done it myself and I know of colleagues in the industry who have also done it. It is more common where you have a regular or semi-regular client who has fallen on hard times or just wants to make life a little easier in some respect. If they have paid on time before, the editor may trust them to pay all their instalments. I have made this arrangement with a 'friend of a friend' - I had only met her once, briefly, but knew our mutual friend (who was also her housemate) quite well and was confident that she would be trustworthy. She paid over four months.
If an editor is prepared to accept payment in instalments it doesn't necessarily mean that you're getting a worse job - it's more about the rapport you have with them and whether they feel they can trust you. For instance, I would probably trust you ;-) because I have seen your post here asking for advice! If I received a 'cold email' saying 'I want my book editing but I can't afford to pay yet' I might think you were a chancer.
If there was an agreement where the full edited book wasn't delivered back to you until payment of the final instalment, that could also be helpful as if you were prepared to accept such an arrangement it would help to prove that you were genuine and protect the editor from the chance of you just scarpering with the work done and never paying. It would also give them longer to work on it, so you might even get a better job in the end!