Is it normal for a new garden fence to shake a little bit in the wind?

hi we recently replaced an old 6ft tall garden fence after strong winds caused the 4x4 inch posts to snap off. we found that the old posts had been cemented in the ground and we could not break up the cement as the whoever put the fence up before us used too much cement around the posts. therefore we replaced the fence using 4x4 inch posts and 6 x 4 foot overlap panels (ones with wooden slats going across) and used those steel post spikes hammered and cemented into the ground to hold the new posts secure. after completion i checked the fence and it felt very secure although i am worrying now after it being up for a few weeks that it may collapse again or something as we have been getting fairly gusty winds lately and i have noticed that the posts and panels shake or wobble a bit when a gust blows across the garden. is this normal and will the fence be okay? thanks :-(