You have to have space on the roster, or you cannot participate in the draft.
If I understand the rules correctly, a club would have to open a space on their roster by outrighting a player (or more than one). In other words, it's not enough to just "designate for assignment" - they actually have to get somebody through outright waivers and outright him (or have him claimed off waivers). This, presumably, is why the rules stipulate the final dates for both requesting outright waivers (Rule 10(c)(3)(B)) prior to the Rule 5 draft, and for actually outrighting players (Rule 9(b)), prior to the draft. See the Key Dates post (pinned) - those dates this year are December 7th and 11th.
The club could presumably clear a roster spot after December 11th by trading somebody. Trades during the Winter Meetings are not exactly unheard of!