The official answer from PayPal is No. But, there are certain situations in which this is possible.

If your PayPal account crosses into a negative balance somehow, you suddenly receive a notice telling you how to make a deposit into your account by Check, Money Order, or Cashier's Check. The help page reads as follows:

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We appreciate your efforts to add funds as soon as possible. Negative balances are prohibited by our User Agreement, and accounts with sustained negative balances may be subject to limited access.

You can restore your balance by mailing a check, money order, or cashier's check:

1. Make the payment payable to PayPal Inc.
2. Write your PayPal-registered name and email address on the payment.
3. Send the payment to:

PayPal Inc.
PO Box 45950
Omaha, NE 68145-0950

Only U.S. currency is accepted for check deposits. If you send a check, it will take 7 days before the funds are credited. If you send a money order, it will be credited 2 days after being deposited in the bank.

You can also add funds by bank account or credit card.

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Unfortunately they also only provide the credit card option when you have a negative balance. While they say that this is only for negative balances, I really doubt they will return something sent via mail.