Yes you can..
Normally Unemployments benefits will be paid from the payroll taxes collected from your employer, irrespective of the payroll taxes are from one employer or multiple employers. It is very complex procedure to determine whether particular claimant is eligible for enemployment benefits or not. Everybody is in a perception that Unemployment benefits will be paid only when applied immediately after layoff (By the bye, unemployment benefits will be paid in case of layoff scenario only, not for firings or voluntarily leaving a company).
The states consider whether claimant worked between 5th quarter and 4th quarter from the claim file date. States consider in terms of quarters. January to March is the 1st Quarter, April to June is the 2nd quarter, July to September is the 3rd quarter adn October to December is the 4th quarter. The maximum amount per week to paid is dependent on each state's rule.
For Example: if the claim file date is 05/01/2003. i.e. the claimant filed for benefits in the 2nd quarter (2nd quarter is April to June and the current month is May). The base period start date is the first day of the 5th quarter before the file date i.e. 1st quarter of previous year. So the Base period start date is 01/01/2002. The base period end date is the last day of the 4th quarter before the quarter in which claim is filed. it is 2rd quarter of the previous year and the date is 06/30/02. Whatever the earnings the claimant earned in this period is considered for Unemployment benefits.
As I said it is a very complex process, but I believe it clears all your doubts. You are eligible for Unemployment benefits for 6 months if you worked between 01/01/2002 and 06/30/2002. Or you are entitled to get benefits based on the payroll taxes collected during this period. You still be eligible for part benefits, you you worked for part of that period.