Yes, you can use multiple class in same field as you did. But there are some mistake in your class file.
it should have '.' prefix in style tag class name like
<style>
.class1
{
color:red;
}
.class2
{
color:blue;
}
.class3
{
color:yellow;
}
</style>
Now, in such case, last class will be effective (mainly the last overlapping property)