It's unlikely to come down but the extra weight of the new plaster on the underlying substrate has a bearing on how solid it will be. You can get settlement cracks, especially in ceilings, because of movement in the floor above with people walking on it. I think it sounds like re-boarding is the best all round solution to me.
Do you know if the original ceiling is plasterboard or if it could be lath and plaster?
If it is the latter, re-boarding over would have been a good idea in the first place because those type of older ceilings can sometimes partially come down if they are not solid, and getting it done now will help hold everything in place. (I wouldn't worry too much, but it is possible.) My concern with this is that you make sure the plasterer screws the new boards up rather than nail them because, if the existing ceiling did loosen later down the line, nails wouldn't be anywhere near as strong as screws to hold everything together.
If the original is plasterboard, there is very little chance of anything caving in. If the cracks are around 4 or 8 feet apart, and are in relatively straight lines, this could suggest that joints in plasterboard have cracked, which is nothing major.