No, people aren't wrong who say God always answers our prayers. It would be wrong, however, to say He always answers them the way we want Him to. Let me explain.
One of the Bible's greatest truths is that God loves us - and because He loves us, He hears our prayers when we come to Him in faith. The Bible says, "While they are still speaking I will hear" (Isaiah 65:24). Just as an earthly father listens to his children, so God - our heavenly Father - hears the prayers of His children.
But God also knows what is best for us - which is why He sometimes says "No" to our requests. We may think He's ignoring us or refusing to hear us - but that isn't true. God always answers our prayer - even when His answer was "No", or "Not yet." We see only a small part of the picture - but God sees the whole, and He knows what is best. The Apostle Paul undoubtedly asked God to release him from prison so he could continue preaching, but God said "No" - and as a result Paul had the time to write some of his most important letters.
Thank God for the privilege of prayer - and thank Him too that He knows what is best for us, even when we don't understand. Thank Him most of all for Jesus Christ, who by His death and resurrection opened heaven's doors for us, enabling us to come to God in prayer.