The definition of "Free Software" from the Free Software Foundation:

“Free software” is a matter of liberty, not price. To understand the concept, you should think of “free” as in “free speech,” not as in “free beer.”

Free software is a matter of the users' freedom to run, copy, distribute, study, change and improve the software. More precisely, it means that the program's users have the four essential freedoms:

• The freedom to run the program, for any purpose (freedom 0).
• The freedom to study how the program works, and change it to make it do what you wish (freedom 1). Access to the source code is a precondition for this.
• The freedom to redistribute copies so you can help your neighbor (freedom 2).
• The freedom to distribute copies of your modified versions to others (freedom 3). By doing this you can give the whole community a chance to benefit from your changes. Access to the source code is a precondition for this.

A program is free software if users have all of these freedoms. Thus, you should be free to redistribute copies, either with or without modifications, either gratis or charging a fee for distribution, to anyone anywhere. Being free to do these things means (among other things) that you do not have to ask or pay for permission to do so.

The definition of "Open Source Software" from the Open Source Initiative:

Open source doesn't just mean access to the source code. The distribution terms of open-source software must comply with the following criteria:

1. Free Redistribution The license shall not restrict any party from selling or giving away the software as a component of an aggregate software distribution containing programs from several different sources. The license shall not require a royalty or other fee for such sale.

2. Source Code The program must include source code, and must allow distribution in source code as well as compiled form. Where some form of a product is not distributed with source code, there must be a well-publicized means of obtaining the source code for no more than a reasonable reproduction cost preferably, downloading via the Internet without charge. The source code must be the preferred form in which a programmer would modify the program. Deliberately obfuscated source code is not allowed. Intermediate forms such as the output of a preprocessor or translator are not allowed.

3. Derived Works The license must allow modifications and derived works, and must allow them to be distributed under the same terms as the license of the original software.

4. Integrity of The Author's Source Code The license may restrict source-code from being distributed in modified form only if the license allows the distribution of "patch files" with the source code for the purpose of modifying the program at build time. The license must explicitly permit distribution of software built from modified source code. The license may require derived works to carry a different name or version number from the original software.

5. No Discrimination Against Persons or Groups The license must not discriminate against any person or group of persons.

6. No Discrimination Against Fields of Endeavor The license must not restrict anyone from making use of the program in a specific field of endeavor. For example, it may not restrict the program from being used in a business, or from being used for genetic research.

7. Distribution of License The rights attached to the program must apply to all to whom the program is redistributed without the need for execution of an additional license by those parties.

8. License Must Not Be Specific to a Product The rights attached to the program must not depend on the program's being part of a particular software distribution. If the program is extracted from that distribution and used or distributed within the terms of the program's license, all parties to whom the program is redistributed should have the same rights as those that are granted in conjunction with the original software distribution.

9. License Must Not Restrict Other Software The license must not place restrictions on other software that is distributed along with the licensed software. For example, the license must not insist that all other programs distributed on the same medium must be open-source software.

10. License Must Be Technology-Neutral No provision of the license may be predicated on any individual technology or style of interface.

These definitions, although they derive from very different ideologies, are broadly compatible, and most Free Software is also Open Source Software and vice versa. I believe, however, that it is possible for this not to be the case: It is possible for software to be Open Source without being Free, or to be Free without being Open Source.

Questions

1. Is my belief correct? Is it possible for software to fall into one camp and not the other?
2. Does any such software actually exist? Please give examples.

Clarification

I've already accepted an answer now, but I seem to have confused a lot of people, so perhaps a clarification is in order. I was not asking about the difference between copyleft (or "viral", though I don't like that term) and non-copyleft ("permissive") licenses. Nor was I asking about your personal idiosyncratic definitions of "Free" and "Open". I was asking about "Free Software as defined by the FSF" and "Open Source Software as defined by the OSI". Are the two always the same? Is it possible to be one without being the other?

And the answer, it seems, is that it's impossible to be Free without being Open, but possible to be Open without being Free. Thank you everyone who actually answered the question.