Does a following consonant cancel the preceding consonant?

Some consonant clusters do drop elements completely, especially in rapid speech; but more often they are all there, but only the last is released.

So in rapid speech asked Dave and ask Dave both may sound the same - they lose the /k/ as well as the first /d/ in the former: /aːsdɛɪv/ in my dialect. In slightly more careful speech, the /k/ is still absent, but the two sound different because there is an unreleased /d/ in the former, resulting in to the geminate /aːsdːɛɪv/.

Slowing down further, the /k/ becomes audible, but again unreleased - the cluster is /k͡d/, which begins with the velar closure of /k/, but the tongue moves before the release, so that the it is /d/ which is released. But the two are still distinct, with a longer interval before the release in "asked Dave".

Finally, in very careful speech, all the consonants will be pronounced (and released) individually.

It's not 'cancelling' in the narrow sense. Basically, it's because of English Phonotactics Constraints.

English cannot have two homorganic plosives (voiced and voiceless) next to each other. Homorganic consonants are the consonants that are articulated from the same place.

/t/ and /d/ are both stops and have the same place of articulation i.e. the alveolar ridge. /t/ is voiceless while /d/ is voiced. It's hard to pronounce a voiced sound right after a voiceless sound because changing from voiceless to voiced consonants requires independent movement of the larynx, which can be very hard to switch on and off in the fraction of a second in normal speech.

Here's the Sagittal section of the /t/:

Here's the Sagittal section of the /d/:

As you can see (the encircled place), switching from one position to another position can be hard at times especially in normal speech. Therefore, English does not have clusters like /gk/, /kg/, /td/, /dt/, /pb/, /bp/ etc.

I need two kilos of shrimp.

In normal speech, when one words ends with /d/ or /t/ and another word begins with the consonant that has the same place of articulation with the preceding one, we usually geminate the following sound or some people articulate the preceding sound partially and then quickly start the following sound.

That's why in words like cupboard, background etc., we kind of skip the preceding consonant (or geminate the following consonant).