Brief Answer:
Yes, few of its side effects may resembles those of asthma

Detailed Answer:
Hi,

Olmesartan is safe drug and mostly does not cause many problems however besides it common side effects (raised potassium which is mostly not significant, dizziness) few of its effect may be sounding like asthma. That is it may cause bronchitis, rhinitis, flue and sinusitis. These symptoms may sound like asthma as it can cause cough and discomfort in throat and chest with fever, though these are not very typical like that of asthma but resembles some of that (it would have been great if you could have explained as what you meant by asthma like symptoms). So if you have any such symptoms, we understand this may be related to this drug.

Most of the times such things gradually resolves with time so you can continue with the drug if the symptoms are not too troublesome. However if its causing you more trouble, or difficulty in breathing you can immediately stop it.
Just want to mention that metoprolol (you mentioned in your drugs list) is a beta blocker and may also cause congestion of the lungs (like asthma) specially in those people who already do have asthma or COPD. This is not common in normal population.
Generally we put any effect to a drug only if nothing else can explain that effect. Another way is if the effect occur after the use of the drug and decreases with stopping the drug.

However if you think, this drug is causing it for you, you can always talk to your doctor to change it as there are many other options available.

Hope I have answered your question. Please let me know if you any further question. I would be glad to answer you question.
Regards,
Dr. Rahman

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