Yes, its possible to have an STI and have no symptoms. If someone is concerned about an STI the best thing to do would be to get an STI test.
A condom would really reduce the chance of something passing, but it also depends if any other type of sex happened, for example oral sex is common and most people do not use condoms and it is possible for STI to pass this way.
If your still with your boyfriend I would recommend that he goes for STI testing.
If you are not having sex with him anymore I would recommend that you go to a clinic and tell them when you last had sex with him and ask when you should be tested. Generally, most STI like gonorrhea and chlamydia will show up on a urine or swab test after 2 weeks if they were passed to someone and STI like HIV and syphilis will show up on a blood test usually by 6 weeks.
Let us know if you have any more questions or concerns.