Just as some people can experience anxiety symptoms when they are not anxious, can you be anxious but not experience symptoms?

Curiously yes: There are non-fearful panic attacks. In some men in particular, they will experience anger attacks instead of panic, even though the underlying emotion is panic. Then there are those people with an inability to read there own emotions (alexithymia), who are anxious but can tell it. Others see it in them, but they do not feel it.

Anxious: Each person has his or her experience of being anxious. Some have a twitchy eye, butterflies in stomach, dry mouth, etc. You can be so disconnected from your inner experience that you don't notice it, though. Part of the work of healing is to re-connect, since your body & emotions can provide a lot of useful information. For most rewarding living, it's best to have all of you in contact.

Yes, it is possible: Anxiety can manifest in several ways: "nervousness", fear, apprehension or what we refer to as psychic anxiety. It can also manifest in a variety of physical symptoms (somatic anxiety ) like palpitations, breathlessness, nausea, gastrointestinal cramps, diarrhea; or as a speech impediment like stuttering. All these can occur without necessarily experiencing the "feeling" of nervousness.