If your question is whether an LLC or partnership can own a NOW account, the answer to both is "No."
However, LLCs and partnerships can have other accounts that are interest-bearing, including a demand deposit account (if your bank offers that product).
The IRS allows a single-member LLC to elect to be ignored as an entity for IRS purposes and use the Taxpayer Identification Number of its owner (an SSN if the owner is an individual). The LLC must obtain an Employer Identification Number (EIN) if it has employees, in order to deposit payroll taxes, and may obtain an EIN if the LLC is required to provide one to open a bank account. As a financial institution, you may accept either number in the case of a single-member LLC that has elected to be ignored as an entity.