There are A-positions where no structural case is assigned. Let me give you some examples from English:
(1) The data were classified
(2) They arrived
(3) They seem to have arrived
(4) The data seem to be classified
(1) is a passive sentence. The complement position (V [DP]) is an A-position where no structural accusative case has been assigned. This position is left without its case (or say the passivized verb does not assign accusative case), this is why the object the data has to move to another position where it ends up assigned a nominative. A similar observation carries over to (2) as an unaccusative verb.
The sentences in (3)/(4) are examples of raising to subject. The specifier position (A-position) of an embedded infinitival clause does not assign a nominative case. Again their subjects have to move further in the structure ending up assigned a nominative by a tensed matrix clause.
Notice that (3) combines two A-positions left without case. The first is the embedded A-position (V-complement) of the unaccusative arrive and the second is the embedded infinitival position (Spec,IP). The same thing with (4), there are two positions left. The embedded passive V-complement and the embedded infinitival Spec,IP.
These are the two structural case positions left without case in a language like English. This observation, however, varies from one language to another.