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Yes you can but be carefull with aluminum house wiring. If you have this type of wiring then call your local inspector to find what wiring techniques are permitted.
NEC Article 314.16 (B)(1) discusses box fill.
"Each conductor that originates outside the box and terminates or is spliced within the box shall be counted once"

"Each conductor that passes throught the box without splice or termination shall be counted once."

So NEC does not specifiy how many times the wire may be spliced.

A good answer above but also bear in mind that any splice must be in a covered, accessible box. If you have to splice the wire outside of the box you'll have to do it in another box in the attic and then run the new pig tail into the light box. If it is between floors you'll have to have another box with an accessible cover in your ceiling. That's not very attractive so hopefully you can do the splice in the light box.

On a note about the previous reply about aluminum wiring: Be very wary if you do have aluminum wiring. It's a source for many house fires. Current code does not allow it's use inside of the home but older homes may have it before the code waas changed. You likely have copper for the small circuits so you shouldn't encounter it with this repair but if you do call an electrician. The splice will need dielectric paste and a special connector. It's not something for a homeowner to risk.

Todd Shell
Todd's Home Services
www.thomeservices.com