Q: I am considering doing the packing myself in order to save some money, when I come on aliya. Is this a good idea?

A: Packing services, like every other service, cost money. If you pack yourself, you should be entitled to a discount.
Usually though, the savings are not that significant. Many times, a client will pack all the books and clothing leaving the breakables and furniture to the professional packing crew. Keep in mind that packing books and clothing goes very quickly for a professional packer - what will take you weeks to pack may take a packing crew only one or two hours. Add to that the time to find boxes for packing or expense of buying them and you will end up saving very little.
One way you will save, by packing yourself, results from your packing "tighter" than the movers. You will fill every nook and cranny in a box by going from room to room to find that perfect sized nick knack to fill a hole. Movers will grab whatever is in hand's reach - they will not walk around to find something to fit in a space.
I once had a client who removed all the of the pot and pan handles. He probably reduced the volume of of his kitchen ware by 50%. A mover will never take time to do that.
You should be aware though - since 9/11, many top moving companies will not accept PBO (Packed By Owner) boxes. They will open and examine every prepacked box. If this is the case, you will receive no savings.
Finally, you should consider the impact on insurance. Boxes packed by owner do not enjoy "All Risk" protection. The insurance companies say that if a box was not professionally packed they cannot be responsible for damage. (If a box is missing, then, you do have coverage).
If you are already packing by yourself, one of the best ways to save money is to bring the packed boxes to the shipping company's warehouse. If your mover does not have to send a truck and crew to your house, the mover's expenses are greatly reduced and you are entitled to that savings.
Bottom line, self-packing is usually not worth it unless you pack everything and bring the boxes to the mover's warehouse. Sometimes, just the idea of packing for weeks vs watching professional packers finish packing everything in a few hours is worth the extra expense.